The Effect of Group Size and Communication Modes in CSCW Environments Masood Masoodian Department of Computer Science University of Waikato , Hamilton , New Zealand masood@cs.waikato.ac.nz Mark Apperley Department of Computer Science University of Waikato , Hamilton , New Zealand m.apperley@cs.waikato.ac.nz Abstract An earlier study -LSB- 7 -RSB- has examined the impact of different human-to-human communication modes on computer supported work involving groups of two people . Based on the findings of this study an experiment was designed to explore the relative impact of face-to-face and audio-only communication modes in supporting shared-workspace interaction between groups of three people . This paper describes the experiment , and examines its findings with the aim of establishing the relationship between the effectiveness of the various communication modes and the group size . 1 . Introduction Over the last two decades the importance of group work , particularly computer supported group work , has been realised by both researchers and commercial organisations . This realisation has in turn led to an increased interest in the study of various aspects of group work . Although there are a large number of factors that affect the process of group work , three of these factors could be considered more important than others . Williams -LSB- 22 -RSB- identifies these factors to be the communication modes available , the nature of the task involved , and the size of the group . There is a growing volume of literature reporting studies whose purpose is to compare various forms of constrained human-to-human communication -LSB- 13 , 17 , 22 , 19 , 12 , 16 -RSB- . These studies have focused on different configurations of text-based , audio , video and face-to - face * meeting environments with the aim of establishing their effectiveness in supporting different types of collaborative group work . There are also studies which have examined a variety of group tasks -LSB- 2 , 15 , 13 -RSB- . McGrath -LSB- 9 -RSB- has given a classification of these tasks together with a list of studies which have utilised them . He identified eight different group task types , which he then categorised into two groups of cooperative and conflict type . The cooperative task category , which includes planning , creative , intellective -LRB- problem-solving -RRB- , and psycho-motor types , represents the tasks that have to be ? solved ? . The conflict task types -LRB- decision-making , cognitive conflict , mixedmotive , and competitive -RRB- on the other hand , are the tasks that have to be ? resolved ? . This may involve resolving conflict of view-points or conflict of interests between the group members . It should be noted that McGrath?s task classification is not the only one available ; it is however derivative of a number of others and is by far the most commonly used task type classification . A review of the current literature on group work and human-to-human communication would show that most of the published research in this area has considered the two major variables of media of communication and communication task , while ignoring the third important factor of group size . In fact the majority of these studies were carried out with groups of just two people -LSB- 2 , 12 , 14 , 15 , 18 -RSB- . However as Williams -LSB- 22 -RSB- suggests , since group processes are more complex in larger groups , it is reasonable to predict that the effects of media differences are more likely to appear as the group size increases . Although in the past few years there have been a number of studies that have concentrated on larger groups of three -LSB- 10 , 13 -RSB- or four -LSB- 3 , 16 -RSB- , there are few that have focused on the group size as the main factor under investigation . One such study has been carried out by Valacich et al. -LSB- 20 -RSB- . This study , which is one of a series , investigates the effects of group size on the process of * The term face-to-face is commonly used in the CSCW literature to describe situations where participants are co-located and in view of one another . idea-generation in computer-based and non-computerbased work environments . It was found that large groups using a computer-based idea-generation system outperformed groups that did not use such system . Groups working in a computer-base environment also outperformed nominal groups -LRB- equivalent numbers of individuals working alone -RRB- . This paper reports on an experiment designed to study the relative effectiveness of audio and face-to-face communication modes in a cooperative shared-workspace problem-solving situation involving groups of three people . The experiment extends and complements an earlier study which examined audio , video , and face-toface communication modes in supporting two person groups . It should be noted that unlike many other similar studies , the work described in this paper focuses mainly on variation in group size as an important aspect of computer supported collaborative work . 2 . Background A recent study -LSB- 7 , 5 , 6 -RSB- investigated the effects of four human-to-human communication modes in a sharedworkspace computer supported cooperative work environment . These modes were : ? face-to-face meetings ? remote meetings supported by an audio link ? remote meetings supported by a full motion video link ? remote meetings supported by a slow motion -LRB- five frames per second -RRB- video link Twelve groups of two people worked in each of these environments on a set of four computer-based jigsaw puzzles , each in a different sequence . Two sets of data were collected ; -LRB- i -RRB- through videotaping the subjects ? upper body area and their workspace during the sessions , and -LRB- ii -RRB- through using a set of three questionnaires . These data were analysed using a number of subjective and objective methods . The results of this study -LSB- 7 -RSB- clearly showed that in a computer-supported shared-workspace problem-solving situation , audio communication is very important . This confirmed the results of Chapanis and Ochsman?s work -LSB- 2 , 12 -RSB- which had examined the differences between various communication modes in supporting collaborative noncomputer-based problem-solving work . Sellen -LSB- 16 -RSB- reported similar results from an experiment which compared three different types of video communication with the normal face-to-face and audio-based communication modes , using conflictive type tasks . This study -LSB- 7 -RSB- also indicated that there was no significant difference in the style of interaction between subjects , or the time taken to solve problems , between any of the audio-based , video-based , or face-to-face communication modes . The study indicated that the only difference between these communication modes was reflected in the users ? perception of the media -LSB- 5 -RSB- . Users felt that video added value , and video-based or face-toface communication seemed more ? natural ? . This result was also in agreement with the findings of other related studies -LSB- 3 , 13 -RSB- . Since this empirical study investigated variations to the medium of communication , the other major factors that could be considered for future extensions to this work were the effects of changing the task type and the group size . As was discussed in Masoodian et al. -LSB- 7 -RSB- , for an experiment of this type , which was designed to study the effects of different communication media in a computersupported cooperative work environment , it is only reasonable to use cooperative tasks . Also , of the four tasks that McGrath -LSB- 9 -RSB- classifies as cooperative , only intellective -LRB- problem-solving -RRB- , planning , or creativity type tasks are suitable for a computer-based work environment -LRB- the psycho-motor type tasks are physical tasks which can not be performed on computers -RRB- . Therefore the variation that can be made to the task type for this kind of study is limited . There is also a certain amount of evidence from other work -LSB- 17 , 22 , 9 -RSB- that the effects of changing the type of cooperative task would have very little effect on the group communication process . Thus there was incentive to first consider changing the group size and studying its effects on the group communication process . The remainder of this paper describes an experiment which extends the earlier work mentioned above . The group size for this experiment was changed from two to three people per group . The task type , however , remained the same . In fact to make it possible to compare the results of this experiment with the previous one -LSB- 7 -RSB- , the same set of four jigsaw puzzles were used . In the previous study jigsaw puzzles proved to be very successful and they encouraged the subjects to work to the best of their ability . Each puzzle had a number of tricky parts which required the subjects to cooperate and communicate with one another ; without this cooperation and communication the puzzles would have been difficult to complete . 3 . Experiment Two different modes of meeting were selected for this study . These were : face-to-face meetings , and physically separated meetings supported by an audio link . Although video is usually considered as a communication mode that falls between audio and face-to-face communication modes -LSB- 7 -RSB- , it was not used in this experiment . There were several reasons for this . First , logic would suggest that in terms of effectiveness , video would come between audio and face-to-face -LSB- 7 -RSB- yet the earlier experiment had detected no difference in performance between all three . Second , recent work by Olson et al. . -LSB- 13 -RSB- shows that , in a similar situation , any observable difference between the audio , video , and face-to-face modes existed only between the audio and the face-to-face communication modes . Third , the physical problems of providing views of multiple participants are more difficult to overcome . However , it was decided that if the results did show any difference in this respect then a further experiment would be carried out to include the video communication mode as well . Both of the communication modes utilised a computerbased shared-workspace system , provided by the Aspects ? -LSB- 4 -RSB- conferencing software and three linked Macintosh ? computers . Aspects software allowed the subjects to work interactively on shared drawing documents . A shared workspace was used because it is an important element of synchronous computer supported cooperative work , and other research -LSB- 21 -RSB- has demonstrated its effectiveness in supporting collaborative group work . It is appreciated that Aspects , like other software of this type has a number of shortcomings in such areas as the interface , responsiveness , and floor control . However , because it was used in both of the two settings , the impact of these shortcomings on the communication between the subjects was reduced . 3.1 . Face-to-face meeting environment As Figure 1 shows , in this environment the subjects sat in front of their computers in a triangular setting facing each other . The subjects were able to see each other?s upper-body areas . To keep the recorded sound quality uniform across the two environments , even in the face-to-face mode the subjects communicated through a headset sound system . This in fact also eliminated any differences that might have existed between the sound quality in the face-to-face and voice-only communication modes . Although this made the communication rather different from the usual face-to-face communication , the subjects had no difficulty in adopting to it and they felt that the interaction remained remarkably natural . A Upper-Body Camera Figure 1 : Face-to-face meeting environment Three video cameras were present in the meeting room . These cameras were placed on top of the computer monitors and they recorded the upper-body area of each of the subjects . There was also a fourth computer in the control room which was linked to the computers of the three subjects and showed a view of their shared workspace . The video signals from the three cameras and the control room computer , plus the spoken dialogue between the subjects , were recorded on a single video tape using a four-quadrant video mixer . This was done so that it would be possible during the analysis to simultaneously observe the view of the upper-body area of each of the subjects and their shared workspace . Figure 2 shows a single frame from one of the video tapes . Figure 2 : A single frame of a recorded session showing the three participants and the shared workspace which they each are viewing 3.2 . Voice-only meeting environment The subjects in the second environment -LRB- Figure 3 -RRB- worked on three linked computers located in separate rooms . Each subject was provided with an audio link to the others via a headset . Once again , there was a video camera placed on the computer monitor in each room which captured the images of the upper-body area of the subject working in that room . As with the face-to-face situation , the images from the three cameras , the control room computer -LRB- showing the shared workspace -RRB- , and the audio conversations between the subjects , were recorded on a single video tape for analysis purposes . Audio Headset A Upper-Body Camera A A Room 2 Room 3Room 1 Figure 3 : Voice-only meeting environment 4 . Design McCarthy and Monk -LSB- 8 -RSB- have identified three experimental designs for assigning subjects to groups and groups to different experimental conditions . These are : between subjects/between groups , within subjects/within groups , and within subjects/between groups . In the between subjects/between groups design each subjects takes part in only one group and each group works in only one of the experimental settings . In the within subjects/within groups design however , each subject is assigned to only one group but each group works in all of the environments . Within subjects/between groups is the most complicated design . In this design the subjects work in all the experimental conditions but they are put into new groups for each condition . For this experiment , within subjects/within groups design was chosen . This design uses fewer subjects and allows them to be able to compare the environments . This design is also more sensitive because it makes it possible to separate the individual differences from the error variance -LSB- 8 -RSB- . However , with this design it is important to make sure that each ordering of the environments is experienced by the same number of groups . Four groups of three people were used for this experiment . All the subjects were native-English-speaking males who volunteered to participate . The subjects were all computer science students and they all had good computer skills . Four of the subjects were postgraduate and eight of them were third or fourth-year undergraduate students . The subjects were assigned to the groups randomly . Table 1 shows the ordering of the environments used in this experiment . Each ordering was replicated twice . Numbers 1 to 4 -LRB- in bold style -RRB- show the sequence in which each group worked in the different environments . Each of these numbers also identifies a task on which the given group worked in that environment . Each group performed two tasks in each environment so that the number of samples taken from the environments could be increased while the number of the subjects remained the same . Therefore overall eight samples were taken from each of the environments . Voice-Only Face-to-FaceMode Group 1 2 3 4 1 1 4 4 2 3 4 2 3 4 3 2 1 3 2 1 Table 1 : Ordering of the environments 5 . Data During this experiment two sets of data were collected . The first set was obtained from three different questionnaires . At the beginning of sessions one and three , when the subjects were introduced to a new environment , they were asked to fill out a questionnaire that recorded their expectations of the environment in which they were about to work -LRB- questions 1.1 to 1.5 D in Table 2 -RRB- . At the end of the second and fourth sessions , when the subjects had finished working in an environment , they were given another questionnaire which recorded their reactions to that environment -LRB- questions 2.1 to 2.9 in Table 2 -RRB- . There was also a third questionnaire which was used at the end of the experiment -LRB- end of session four -RRB- to get the subjects ? ranking of the two environments -LRB- questions 3.1 to 3.5 in Table 2 -RRB- . The second type of collected data was the set of video tapes which contained the recorded images of the upperbody area of the three subjects in each group and the image of their shared workspace , plus the audio conversation between the subjects . The video and audio signals were passed through a four channel video mixer and an audio mixer and were recorded on a single video tape . This was done so that the audio and video signals were synchronised for later analysis . 6 . Results The collected data from this experiment were analysed in terms of the style of the conversation between the subjects , the time taken to complete the tasks in each of the environments , and the subjects ? response to the questionnaires . The next two sections describe the results of these analyses in detail . 6.1 . Interaction style and session duration The analysis method for the style of conversation -LSB- 14 , 15 -RSB- is very similar to the one used in the previous study -LSB- 7 -RSB- . However , since there were three subjects in each group who communicated with one another -LRB- rather than the previous dyadic conversation -RRB- , the method had to be changed slightly to include a number of additional variables . The following factors were measured : total speech duration , number of utterances , total duration of simultaneous speech involving two people , total duration of simultaneous speech involving three people , number of simultaneous speech utterances involving two people , number of simultaneous speech utterances involving three people , total duration of mutual silence , number of floor control changes , and the number of unsuccessful attempts to gain the floor control . No predictions were made about the value of these variables in this experiment and they were compared only across the two different environments . Question Answer 1.1 - How satisfied do you think you will be with this communication mode ? -LRB- 1 -RRB- Very Dissatisfied to -LRB- 7 -RRB- Very Satisfied 1.2 - How easy do think it will be to communicate in this environment ? -LRB- 1 -RRB- Very Difficult to -LRB- 7 -RRB- Very Easy 1.3 - How efficiently do you think you can work in this environment ? -LRB- 1 -RRB- Not Very Efficiently to -LRB- 7 -RRB- Very Efficiently 1.4 - How enjoyable do think it will be to work in this environment ? -LRB- 1 -RRB- Not Very Enjoyable to -LRB- 7 -RRB- Very Enjoyable 1.5 - Indicate your expectations about how it would be to work in this environment : 1.5 A -LRB- 1 -RRB- Very Impersonal to -LRB- 7 -RRB- Very Friendly 1.5 B -LRB- 1 -RRB- Very Frustrating to -LRB- 7 -RRB- Not Very Frustrating 1.5 C -LRB- 1 -RRB- Very Time Wasting to -LRB- 7 -RRB- Very Time Saving 1.5 D -LRB- 1 -RRB- Very Unproductive to -LRB- 7 -RRB- Very Productive 2.1 - How satisfied are you with this communication mode ? -LRB- 1 -RRB- Very Dissatisfied to -LRB- 7 -RRB- Very Satisfied 2.2 - How much did you enjoy working on the last two tasks ? -LRB- 1 -RRB- Very Little to -LRB- 7 -RRB- Very Much 2.3 - How efficient was the group work over the last two sessions ? -LRB- 1 -RRB- Not Very Efficient to -LRB- 7 -RRB- Very Efficient 2.1 - How easy was it to control the communication in this environment ? -LRB- 1 -RRB- Very Difficult to -LRB- 7 -RRB- Very Easy 2.5 - How satisfied are you with the group work you did in the last two sessions ? -LRB- 1 -RRB- Very Dissatisfied to -LRB- 7 -RRB- Very Satisfied 2.6 - How easy was it to understand your partners ? ideas in this environment ? -LRB- 1 -RRB- Very Difficult to -LRB- 7 -RRB- Very Easy 2.7 - How easy was it to get your own ideas across in this environment ? -LRB- 1 -RRB- Very Difficult to -LRB- 7 -RRB- Very Easy 2.8 - How much did the other participants contribute to the group work in the last two sessions ? -LRB- 1 -RRB- Very Little to -LRB- 7 -RRB- Very Much 2.9 - How much did you contribute to the group work in the last two sessions ? -LRB- 1 -RRB- Very Little to -LRB- 7 -RRB- Very Much 3.1 - How enjoyable was it to work in the two environments ? -LRB- 1 -RRB- Most Enjoyable to -LRB- 2 -RRB- Least Enjoyable 3.2 - How efficiently did you work in the two environments ? -LRB- 1 -RRB- Most Efficient to -LRB- 2 -RRB- Least Efficient 3.3 - How easy was it to communicate in the two environment ? -LRB- 1 -RRB- Easiest to -LRB- 2 -RRB- Hardest 3.4 - How easy was it to work in the two environments ? -LRB- 1 -RRB- Easiest to -LRB- 2 -RRB- Hardest 3.5 - How would you rank the two environments overall in terms of your own preference ? -LRB- 1 -RRB- Best to -LRB- 2 -RRB- Worst Table 2 : A summary of the three questionnaires The variables were scored directly from the recorded video tapes of the sessions by an observer , using a computer program specifically written for this purpose . The recorded video tapes of the sessions were viewed and depending on whether a subject was talking or not at a given time , a key on the computer keyboard assigned to that subject was pressed or released . Based on these input values the program generated a series of on/off patterns for each of the group members through the entire session . The same observer viewed all the video tapes session by session -LRB- rather than group by group , i.e. first sessions for all the groups were viewed before the second sessions and so on -RRB- so that the accuracy of the logging remained similar between the groups The conversation patterns generated by the first program were then passed through another custom-made program which analysed them and produced a value for each of the measured variables . This program was based on the approach used by Argyle and Cook -LSB- 1 -RSB- for measuring gaze and speech patterns . However , the method had to be extended considerably to be used for analysing 3-way conversations . The final method is very similar to the one used by Sellen -LSB- 16 -RSB- . The session duration was also measured from the video tapes . Only one of the sixteen sessions did not produced a completed jigsaw puzzle . To avoid having a missing value for this session in the analysis process , it was assumed that the session would have taken a long time to complete . A ratio of maximum allowed time over actual session duration was used to compare the time taken to complete the tasks in two environments so that the value used for the incomplete session would be zero . The maximum allowed time was 40 minutes for the third task , and 30 minutes for the other three tasks -LRB- these tasks were a little simpler -RRB- . Table 3 shows a summary of the results of a two-way analysis of variance test -LRB- with multiple observations per cell -RRB- for both the style of interaction and the session duration for the two environments . Since the duration of the sessions were all different , the values of the variables were normalised to a total of 1800 seconds which was the average session duration . As the results show , neither of the measured variables were affected by the changes made to medium of communication . In fact only one of the ten variables -LRB- number of floor changes -RRB- gets close to a significant level . This result is entirely consistent with the findings of the earlier study -LSB- 7 -RSB- , highlighting the fact the style of interaction and the group performance do not seem to be affected by variation in the mode of communication or in the group size . However , the results of this analysis indicated that there was a difference between the four groups -LRB- Table 4 -RRB- for all the variables measured except the session duration . This finding , which is also consistent with the previous study -LSB- 7 -RSB- , indicates that the variation among the groups is greater than the variations among the environments . This is due to the fact that different groups are naturally different in terms of the amount of interaction between their members , or the abilities of their members in solving problems . Face-to-Face Voice-Only F P Mean Mean d ? = 1,8 Maximum Session Duration/Session Duration 1.45 1.42 0.01 0.912 Speech Duration -LRB- s -RRB- 1098 1114 0.03 0.875 Number of Utterances 380 406 1.00 0.346 Number of Simultaneous Speeches Involving 3 People 15 16 0.03 0.872 Simultaneous Speech Duration Involving 3 People -LRB- s -RRB- 19 20 0.00 0.966 Number of Simultaneous Speeches Involving 2 People 136 133 0.02 0.894 Simultaneous Speech Duration Involving 2 People -LRB- s -RRB- 192 176 0.32 0.585 Mutual Silence Duration -LRB- s -RRB- 933 901 0.28 0.609 Number of Floor Changes 218 242 4.77 0.061 Number of Floor Change Attempts 78 75 0.04 0.843 Table 3 : Summary of two-way ANOVA of interaction style and session duration for media Group 1 Group 2 Group 3 Group 4 F P Mean Mean Mean Mean d ? = 3,8 Maximum Session Duration/Session Duration 1.59 1.54 1.22 1.40 0.40 0.755 Speech Duration -LRB- s -RRB- 716 987 1839 883 25.72 0.000 Number of Utterances 253 406 605 309 34.48 0.000 Number of Simultaneous Speeches Involving 3 People 8 9 39 7 6.65 0.014 Simultaneous Speech Duration Involving 3 People -LRB- s -RRB- 9 11 50 8 5.77 0.021 Number of Simultaneous Speeches Involving 2 People 78 102 272 86 23.22 0.000 Simultaneous Speech Duration Involving 2 People -LRB- s -RRB- 106 134 377 119 21.88 0.000 Mutual Silence Duration -LRB- s -RRB- 1208 969 439 1052 31.53 0.000 Number of Floor Changes 144 245 361 170 79.24 0.000 Number of Floor Change Attempts 48 62 152 45 8.72 0.007 Table 4 : Summary of two-way ANOVA of interaction style and session duration for groups Question Size for Test W + W - Estimated Median P 1.1 8 23.0 13.0 0.250 0.529 1.2 9 28.5 16.5 0.500 0.515 1.3 9 32.0 13.0 0.500 0.286 1.4 7 21.0 7.0 0.500 0.272 1.5 A 9 41.5 3.5 1.000 0.028 1.5 B 7 25.0 3.0 1.000 0.076 1.5 C 7 22.0 6.0 0.500 0.205 1.5 D 4 7.5 2.5 0.000 0.465 2.1 6 11.0 10.0 0.000 1.000 2.2 3 2.0 4.0 0.000 0.789 2.3 11 43.0 23.0 0.500 0.398 2.4 10 18.0 37.0 -0.500 0.359 2.5 7 16.0 12.0 0.000 0.800 2.6 7 17.0 11.0 0.500 0.673 2.7 7 17.5 10.5 0.500 0.612 2.8 1 1.0 0.0 0.000 1.000 2.9 6 9.0 12.0 0.000 0.834 Table 5 : Summary of the Wilcoxon Signed Rank Test for questionnaires one and two Question Below Equal Above Median P 3.1 5 7 1.000 0.016 3.2 2 5 5 0.000 0.453 3.3 1 7 4 0.000 0.375 3.4 1 8 3 0.000 0.625 3.5 3 2 7 1.000 0.344 Table 6 : Summary of the Sign Test for questionnaire three 6.2 . Questionnaires The analysis of the questionnaires was reasonably simple . The subjects ? answers to the questions were truncated to the nearest whole number . For questionnaires one and two -LRB- expectations and reactions -RRB- the Wilcoxon signed rank test -LSB- 11 -RSB- was used to compare the values subjects gave to different questions in the two environments . The result of this analysis is shown in Table 5 . The W + and W - values in Table 5 show the sum of the ranks with a positive or negative sign respectively . The Size for Test value on the other hand shows the number of answers which were different for the two environments -LRB- note that there were 12 pairs of answers for each question -RRB- . As the result show , the only question for which the subjects ? answers for the two environments were different to a significant level was question 1.5 A . This means that the subjects expected that the audio-based communication mode would be less friendlier than the face-to-face mode . Once again this finding is completely consistent with the result of the earlier study -LSB- 5 -RSB- , indicating that increasing the group size from two to three had no real effect in the subjects ? expectations or reactions to the audio-based and face-to-face communication modes . Questionnaire three , which was used for obtaining the subjects ? ranking of the two environments , was analysed in a slightly different manner using the sign test -LSB- 11 -RSB- . The result of this test which is shown in Table 6 indicates that only for question 3.1 did the subjects rank the face-to-face environment differently to the audio-based environment . This means that the subjects felt that working in the faceto-face mode was more enjoyable than working in the voice-only mode . Note that the values for Below , Equal , and Above show the number of people who felt that the face-to-face mode was worse , the same , or better than the voice-only environment respectively . Although the result of this questionnaire is different from the result of a similar questionnaire used in the previous experiment -LSB- 5 -RSB- , it is reasonable to say that because there were fewer environments in the current experiment the subjects were able to make a better comparison of them . It should also be noted that even though the face-to-face and voice-only environments were ranked differently in the previous experiment , the subjects indicated that there was very little difference between them . 7 . Conclusions The results of the data analysis discussed in this paper along with the results of the previous work -LSB- 5 , 6 , 7 -RSB- , indicate that an audio channel is an important and sufficient means of communication for supporting sharedworkspace collaboration in a problem-solving situation . Furthermore , these results also show that there is very little change , if any , produced by shifting from audio-only to a face-to-face situation , apart from the more ? natural ? feeling some participants felt in the latter environment . On a communication bandwidth scale , video-based communication would come somewhere between these two -LSB- 5 , 6 , 7 -RSB- so it can be concluded that there would be no significant advantage of video over audio with a sharedworkspace problem-solving task . What is even more interesting is that , for this type of task , changing the group size from two to three seems to have no effect on the measured factors . It should however be noted that since groups of size four or more are once again different from groups of size three -LSB- 13 -RSB- , similar studies with larger groups may give different results . In summary , the current series of empirical studies reported here show that for small collaborative problemsolving groups an audio channel plus a computer supported shared-workspace is sufficient for satisfactory and productive group work . Therefore , rather than attempting to integrate video and other means of communication with audio to support remote sharedworkspace collaboration , industry should be devoting more attention to improving groupware technology which is used to provide the shared-workspace facility for remote collaborative work . 8 . References 1 . Argyle , M. and Cook , M. , Gaze and Mutual Gaze . Cambridge University Press , 1976 . 2 . Chapanis , A. , Interactive Human Communication . Scientific American , 232 , -LRB- 3 -RRB- , 1975 , 36-42 . 3 . 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Tang , J. C. , and Isaacs , E. , Why Do Users Like Video ? Studies of Multimedia-Supported Collaboration . Computer Supported Cooperative Work An International Journal -LRB- CSCW -RRB- , 1 , -LRB- 3 -RRB- , 1993 , 163-196 . 20 . Valacich , J. S. , Dennis , A. R. , and Connolly , T. , Idea Generation in Computer-Based Groups : A New Ending to an Old Story . Organizational Behavior and Human Decision Processes , 57 , 1994 , 448-467 . 21 . Whittaker , S. , Geelhoed , E. , and Robinson , E. , Shared Workspaces : How Do They Work and When Are They Useful ? International Journal of Man-Machine Studies , 39 , -LRB- 5 -RRB- , 1993 , 813-842 . 22 . Williams , E. , Experimental Comparison of Face-to-Face and Mediated Communication : A Review . Psychological Bulletin , 84 , -LRB- 5 -RRB- , 1977 , 963-976 .

O O O B I O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I I I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

communication mode
computer supported cooperative work
cscw
group communication
group size
group work
shared workspace


AALBORG

UNIVERSITY

Optimal

Junction

Trees

by

Finn

V.

Jensen

Frank

Jensen

R

ISSN

January

Institute

for

Electronic

Systems

Department

of

Mathematics

and

Computer

Science

Fredrik

Bajers

Vej

DK

Aalborg

O

Denmark

Tel.

TELEX

aub

dk

d

Optimal

Junction

Trees

Finn

V.

Jensen

Frank

Jensen

Department

of

Mathematics

and

Computer

Science

Aalborg

University

Fredrik

Bajers

Vej

Aalborg

Ost

Denmark

E-mail

fvj@iesd.auc.dk,

fj@iesd.auc.dk

Abstract

The

paper

deals

with

updating

of

beliefs

in

networks

First

it

is

shown

that

any

exact

propagation

scheme

must

contain

a

triangulation

of

the

network

and

therefore

essentially

is

a

message

passing

scheme

over

a

junction

tree

In

the

second

part

of

the

paper

we

give

a

simple

algorithm

for

constructing

an

optimal

junction

tree

from

a

triangulated

network

Introduction

Several

algorithms

for

probability

updating

in

Bayesian

networks

-LRB-

and

Markov

networks

-RRB-

exist

Shachter

et

al.

-LRB-

-RRB-

show

the

equivalence

of

several

of

them

They

showed

that

the

methods

known

at

that

time

in

fact

can

be

viewed

as

performing

calculations

on

a

secondary

structure

derived

from

a

triangulation

of

the

network

This

work

has

been

partially

funded

by

the

Danish

research

councils

through

the

PIFT

programme

Still

new

methods

are

proposed

e.g.

-LRB-

Zhang

and

Poole

-RRB-

In

the

first

part

of

this

paper

we

show

that

any

exact

method

in

fact

must

contain

a

triangulation

of

the

network

and

in

principle

will

contain

a

message

passing

in

junction

trees

Our

arguments

are

not

only

valid

for

probability

updating

They

include

all

quantitative

schemes

with

propositional

calculus

as

a

special

case

-LRB-

like

fuzzy

logic

and

DempsterShafer

belief

functions

-RRB-

The

junction

tree

propagation

method

-LRB-

Jensen

et

al.

Lauritzen

and

Spiegelhalter

-RRB-

is

designed

for

propagation

in

Markov

networks

ffl

an

undirected

graph

with

discrete

variables

as

nodes

ffl

for

each

clique

U

in

the

graph

there

is

a

potential

OEU

which

is

a

non-vanishing

function

from

the

set

of

configurations

of

U

to

the

set

of

non-negative

reals

The

compilation

part

of

the

method

is

to

ffl

triangulate

the

graph

-LRB-

i.e.

add

extra

links

such

that

every

cycle

of

length

greater

than

three

has

a

chord

-RRB-

ffl

form

a

potential

OEU

for

each

clique

U

of

the

triangulated

graph

ffl

construct

a

junction

tree

over

the

cliques

A

junction

tree

over

the

cliques

is

characterized

by

the

so-called

junction

tree

property

For

each

pair

U

V

of

cliques

with

intersection

S

all

cliques

on

the

path

between

U

and

V

contain

S

The

propagation

part

of

the

method

consists

of

ffl

giving

all

links

in

the

junction

tree

a

label

consisting

of

the

intersection

of

the

adjacent

nodes

these

labels

are

called

separators

-LRB-

see

Figure

-RRB-

ffl

attaching

a

potential

to

all

separators

-LRB-

initially

the

neutral

potential

consisting

of

ones

-RRB-

ffl

letting

the

nodes

communicate

via

the

separators

a

message

from

U

to

V

with

separator

S

has

the

form

that

OEU

is

marginalized

down

to

S

resulting

in

OE

S

OE

S

is

placed

on

the

separator

and

OE

-LRB-

S

-RRB-

OE

-LRB-

S

-RRB-

is

multiplied

on

OEV

-LRB-

see

Figure

-RRB-

ABCABABDAAAHAEFAEAAEGAIUSV

-LRB-

a

-RRB-

-LRB-

b

-RRB-

-LRB-

S

-RRB-

-LRB-

S

-RRB-

-LRB-

S

-RRB-

Figure

-LRB-

a

-RRB-

A

junction

tree

-LRB-

b

-RRB-

Message

passing

in

junction

trees

It

is

so

that

after

a

finite

number

of

message

passes

between

neighbours

in

the

junction

tree

each

potential

in

the

junction

tree

holds

the

-LRB-

possibly

non-normalized

-RRB-

marginal

of

the

joint

probability

distribution

for

the

entire

set

of

variables

In

fact

the

message

passing

can

be

organized

so

that

it

is

sufficient

with

exactly

one

pass

in

each

direction

of

the

links

in

the

junction

tree

Therefore

in

complexity

considerations

for

propagation

in

junction

trees

one

can

associate

a

local

measure

C

-LRB-

U

V

-RRB-

to

links

-LRB-

U

V

-RRB-

where

C

-LRB-

U

V

-RRB-

indicates

time/space

consumption

for

the

two

passes

The

compilation

is

not

deterministic

Markov

networks

may

have

several

different

triangulations

yielding

different

sets

of

cliques

and

a

triangulated

network

may

have

several

different

junction

trees

We

therefore

would

like

to

have

algorithms

yielding

optimal

triangulations

and

optimal

junction

trees

with

respect

to

complexity

The

optimality

problem

for

triangulations

is

NP-hard

and

we

shall

not

deal

with

this

problem

here

In

the

second

part

of

the

paper

we

address

the

optimality

problem

for

junction

trees

given

the

triangulated

graph

and

we

present

a

simple

algorithm

which

is

quadratic

in

the

number

of

cliques

The

necessity

of

triangulation

The

situation

in

Markov

chain

propagations

is

that

a

universe

U

fA

Bg

of

discrete

variables

is

given

The

joint

probability

P

-LRB-

U

-RRB-

is

a

distribution

over

the

configurations

XU

A

B

The

specification

of

P

-LRB-

U

-RRB-

consists

of

a

set

fP

-LRB-

-RRB-

P

-LRB-

Un

-RRB-

g

of

distributions

on

sets

Ui

U

We

may

assume

that

initially

P

-LRB-

Ui

-RRB-

is

the

marginal

of

P

-LRB-

U

-RRB-

to

Ui

and

we

may

also

assume

that

Ui

n

Uj

for

i

j

The

propagation

task

can

be

formulated

as

follows

Let

P

-LRB-

-RRB-

be

substituted

for

P

-LRB-

-RRB-

If

P

-LRB-

-RRB-

P

-LRB-

-RRB-

P

-LRB-

Un

-RRB-

determine

a

distribution

P

-LRB-

U

-RRB-

uniquely

calculate

the

new

marginals

P

-LRB-

-RRB-

P

-LRB-

Un

-RRB-

By

a

local

propagation

scheme

we

understand

a

solution

of

the

propagation

task

through

a

fixed

set

Prm

of

procedures

where

each

procedure

Pri

can

process

information

on

a

fixed

set

Vi

of

variables

We

call

Vi

the

scope

of

Pri

Our

claim

is

that

any

local

propagation

scheme

is

essentially

the

result

of

a

triangulation

First

however

we

shall

generalize

the

concept

of

general

propagation

task

a

bit

and

also

make

it

more

precise

As

mentioned

earlier

the

propagation

task

is

not

only

a

task

in

probability

updating

We

shall

deal

with

all

types

of

potentials

which

have

propositional

calculus

as

a

special

case

In

propositional

calculus

the

potentials

on

sets

of

variables

are

tables

of

possible

configurations

Since

a

configuration

is

possible

if

and

only

if

its

probability

is

positive

we

see

that

propositional

calculus

is

a

special

case

of

probability

calculus

In

order

to

be

able

to

formulate

the

general

propagation

task

we

also

require

that

the

potential

type

has

a

projection

operator

which

uniquely

determines

potentials

on

subsets

Marginalization

is

the

projection

in

probability

calculus

and

in

propositional

calculus

the

projection

is

performed

through

maximization

In

general

we

require

that

if

V

W

then

projection

to

W

followed

by

projection

to

V

is

the

same

as

a

projection

directly

to

V

We

shall

call

the

types

of

potentials

meeting

these

two

requirements

admissible

A

scene

for

a

general

propagation

task

consists

of

a

universe

U

of

-LRB-

discrete

-RRB-

variables

and

a

covering

Un

of

U

The

scene

is

characterized

by

an

undirected

graph

G

with

U

as

node

set

and

links

between

variables

being

members

of

the

same

Ui

in

the

covering

We

shall

represent

a

scene

by

its

characterizing

graph

The

cliques

of

G

forms

the

covering

of

U.

A

general

propagation

task

is

a

pair

-LRB-

G

R

-RRB-

where

G

is

a

graph

and

R

is

an

admissible

potential

type

An

instance

of

a

general

propagation

task

-LRB-

G

R

-RRB-

is

a

potential

R

from

R

over

the

node

set

A

local

propagation

scheme

for

a

general

propagation

task

-LRB-

G

R

-RRB-

is

a

set

Prm

of

procedures

of

fixed

scope

solving

the

propagation

task

for

all

instances

of

-LRB-

G

R

-RRB-

Theorem

Let

-LRB-

G

R

-RRB-

be

a

general

propagation

task

and

let

Prm

be

a

local

propagation

scheme

for

-LRB-

G

R

-RRB-

Let

Vm

be

the

scopes

of

Prm

and

let

G

be

the

graph

with

the

universe

as

node

set

and

with

links

between

variables

in

the

same

scope

Then

G

contains

a

triangulation

of

G.

the

cliques

are

not

the

covering

then

they

are

unions

of

Uis

In

that

case

we

need

a

combination

operator

for

the

potentials

This

is

however

not

important

for

our

considerations

BACD

Figure

A

graph

representing

a

general

propagation

task

Example

Let

the

graph

in

Figure

represent

a

general

propagation

task

over

the

propositional

calculus

and

let

Pos

be

the

potential

giving

for

possible

configurations

and

for

impossible

ones

Let

PrAB

PrAC

PrBD

PrDC

be

procedures

for

solving

the

task

-LRB-

the

index

indicates

the

scope

see

Figure

-RRB-

We

shall

construct

an

instance

which

can

not

be

solved

by

the

procedures

For

each

variable

we

only

use

the

first

two

states

This

means

that

all

other

states

are

impossible

Initially

we

have

Pos

-LRB-

ai

bj

-RRB-

for

all

i

j

Pos

-LRB-

ai

cj

-RRB-

if

and

only

if

i

j

Pos

-LRB-

bi

dj

-RRB-

if

and

only

if

i

j

Pos

-LRB-

ci

dj

-RRB-

for

all

i

j

That

is

A

and

C

as

well

as

B

and

D

are

forced

into

the

same

state

and

everything

else

is

possible

Note

that

the

Pos-relations

above

are

projections

of

a

unique

Pos-relation

over

the

universe

Pos

-LRB-

ai

bj

ck

d

-RRB-

if

and

only

if

Pos

-LRB-

ai

bj

-RRB-

Pos

-LRB-

ai

ck

-RRB-

Pos

-LRB-

bj

d

-RRB-

Pos

-LRB-

ck

d

-RRB-

ABAACCDCDBDB

Figure

The

scopes

for

the

procedures

and

the

communication

channels

Now

assume

we

get

the

information

that

the

configurations

-LRB-

-RRB-

and

-LRB-

-RRB-

are

impossible

This

is

equivalent

to

the

replacement

of

the

relation

Pos

-LRB-

ai

bj

-RRB-

by

Pos

-LRB-

ai

bj

-RRB-

if

and

only

if

i

j

-LRB-

and

i

j

-RRB-

Now

the

propagation

task

is

to

determine

Pos

-LRB-

A

C

-RRB-

Pos

-LRB-

B

D

-RRB-

and

Pos

-LRB-

C

D

-RRB-

such

that

these

local

relations

are

projections

of

the

unique

universal

relation

Pos

-LRB-

A

B

C

D

-RRB-

satisfying

the

relations

Pos

-LRB-

A

B

-RRB-

Pos

-LRB-

A

C

-RRB-

Pos

-LRB-

B

D

-RRB-

and

Pos

-LRB-

C

D

-RRB-

Clearly

Pos

-LRB-

ai

bj

ck

d

-RRB-

if

and

only

if

i

j

k

and

therefore

Pos

-LRB-

ck

d

-RRB-

if

and

only

if

k

The

tool

for

achieving

this

result

is

the

set

PrAB

PrAC

PrBD

and

PrCD

of

procedures

Since

PrAB

can

only

process

information

on

the

variables

A

and

B

and

PrAC

can

only

process

information

on

A

and

C

then

the

only

valuable

information

to

communicate

between

the

two

procedures

is

information

on

A

-LRB-

see

Figure

-RRB-

That

is

between

and

with

scopes

and

respectively

only

information

on

need

to

be

communicated

The

new

relation

Pos

-LRB-

A

B

-RRB-

introduces

a

constraint

between

the

state

of

A

and

the

state

of

B

but

since

only

information

on

A

alone

and

B

alone

can

be

communicated

the

constraint

can

not

be

communicated

to

PrCD

Note

that

if

a

cycle

contains

more

than

variables

the

construction

can

be

extended

by

clamping

the

states

of

further

intermediate

variables

Proof

of

Theorem

Assume

that

G

does

not

contain

a

triangulation

of

G

Then

there

is

a

cycle

C

in

G

such

that

the

subgraph

of

G

consisting

of

the

nodes

in

C

is

not

triangulated

Let

C

be

a

chordless

cycle

of

length

greater

than

three

in

that

subgraph

Let

An

be

the

nodes

of

C

We

now

can

construct

an

instantiation

which

can

not

be

propagated

correctly

-LRB-

-RRB-

Let

a

configuration

be

possible

if

and

only

if

its

projection

to

An

is

possible

-LRB-

-RRB-

Perform

the

construction

as

shown

in

the

example

Junction

trees

and

maximal

spanning

trees

Throughout

the

remainder

of

the

paper

we

consider

a

triangulated

connected

graph

G

with

clique

set

C

The

cliques

of

G

are

denoted

by

the

letters

U

V

W

U

etc.

We

shall

not

distinguish

between

a

clique

and

its

set

of

variables

So

we

talk

of

the

intersection

of

cliques

meaning

the

set

of

variables

common

to

the

cliques

Intersections

are

denoted

by

letters

R

S

R

etc.

Definition

The

junction

graph

for

G

has

C

as

nodes

and

for

each

pair

U

V

of

cliques

with

nonempty

intersection

R

there

is

a

link

with

label

R.

Each

link

has

a

weight

which

is

the

number

of

variables

in

the

label

Theorem

A

spanning

tree

for

the

junction

graph

of

G

is

a

junction

tree

if

and

only

if

it

is

a

spanning

tree

of

maximal

weight

Theorem

has

been

proved

independently

by

Shibata

-LRB-

-RRB-

and

Jensen

-LRB-

-RRB-

Here

we

will

give

a

much

simpler

proof

than

the

original

ones

Before

giving

the

proof

we

shall

recall

two

algorithms

for

the

construction

of

maximal

spanning

trees

Algorithm

-LRB-

Prim

-RRB-

-LRB-

-RRB-

Put

N

fUg

where

U

is

an

arbitrary

node

-LRB-

-RRB-

Choose

successively

a

link

-LRB-

W

V

-RRB-

of

maximal

weight

such

that

W

N

and

V

N

and

add

V

to

N.

Prim

's

algorithm

constructs

a

sequence

Tn

of

maximal

spanning

trees

for

the

subgraph

determined

by

N.

Algorithm

-LRB-

Kruskal

-RRB-

Choose

successively

a

link

of

maximal

weight

not

producing

a

cycle

Kruskal

's

algorithm

works

with

a

forest

of

partial

maximal

weight

spanning

trees

Whenever

a

link

is

chosen

two

partial

trees

are

connected

into

a

new

partial

spanning

tree

of

maximal

weight

Both

algorithms

result

in

maximal

weight

spanning

trees

and

each

maximal

weight

spanning

tree

can

be

constructed

through

both

algorithms

-LRB-

Proofs

can

be

found

in

any

good

textbook

on

graph

algorithms

-RRB-

Proof

of

Theorem

Let

T

be

a

spanning

tree

of

maximal

weight

Let

it

be

constructed

by

Prim

's

algorithm

such

that

Tn

T

is

a

sequence

of

partial

maximal

weight

spanning

trees

Assume

that

T

is

not

a

junction

tree

Then

at

some

stage

m

we

have

that

Tm

can

be

extended

to

a

junction

tree

T

while

Tm

can

not

Let

-LRB-

U

V

-RRB-

with

label

S

be

the

link

chosen

at

this

stage

V

Tm

-LRB-

see

Figure

-RRB-

Since

Tm

can

not

be

extended

to

a

junction

tree

the

link

-LRB-

U

V

-RRB-

is

not

a

link

in

T

So

there

is

a

path

in

T

between

U

and

V

not

containing

-LRB-

U

V

-RRB-

This

path

must

contain

a

link

-LRB-

U

V

-RRB-

with

label

S

such

that

U

Tm

and

V

Tm

-LRB-

see

Figure

-RRB-

Since

T

is

a

junction

tree

we

must

have

S

S

and

since

S

was

chosen

through

Prim

's

algorithm

at

this

stage

we

also

have

jSj

jS

j.

Hence

S

S

U

'S

V'USV

Figure

Paths

in

T

and

T

Now

remove

the

link

-LRB-

U

V

-RRB-

from

T

and

add

the

link

-LRB-

U

V

-RRB-

The

result

is

a

junction

tree

extending

Tm

contradicting

the

assumption

that

it

can

not

be

extended

to

a

junction

tree

Next

let

T

be

any

non-maximal

spanning

tree

We

shall

prove

that

T

is

not

a

junction

tree

Again

let

T

be

a

sequence

of

maximal

trees

constructed

through

Prim

's

algorithm

Let

the

construction

be

so

that

a

link

from

T

is

chosen

whenever

possible

Let

m

be

the

first

stage

where

this

is

not

possible

and

let

-LRB-

U

V

-RRB-

with

separator

S

be

the

link

actually

chosen

-LRB-

U

Tm

V

Tm

-RRB-

In

T

there

is

a

path

between

U

and

V

As

in

the

first

part

of

the

proof

we

have

that

this

path

contains

a

link

-LRB-

U

V

-RRB-

with

label

S

such

that

U

Tm

and

V

Tm

-LRB-

see

Figure

-RRB-

Since

-LRB-

U

V

-RRB-

could

not

be

chosen

we

have

jS

j

jSj

and

therefore

S

contains

variables

not

in

S

Hence

T

does

not

satisfy

the

junction

tree

condition

Optimal

junction

trees

Whenever

the

junction

graph

has

several

spanning

trees

of

maximal

weight

there

are

accordingly

several

junction

trees

Assume

that

there

is

a

real-valued

measure

on

junction

trees

yielding

a

priority

among

them

and

assume

that

this

measure

can

be

decomposed

to

a

local

measure

D

-LRB-

U

V

-RRB-

attached

to

the

links

We

call

the

measure

a

distance

but

it

may

also

be

a

measure

of

computational

complexity

We

may

also

assume

that

the

entire

measure

is

the

sum

of

the

local

measures

and

that

an

optimal

junction

tree

is

one

of

maximal

distance

-LRB-

the

case

of

minimal

distance

is

completely

analogous

-RRB-

Let

us

take

a

closer

look

at

the

construction

of

junction

trees

through

Kruskal

's

algorithm

Let

wn

be

the

different

weights

of

G

in

decreasing

order

The

algorithm

can

be

considered

as

running

through

n

stages

characterized

by

the

weight

of

the

links

chosen

At

the

end

of

stage

i

all

links

possible

of

weight

wi

have

been

chosen

and

a

forest

Tiki

of

partial

maximal

weight

spanning

trees

has

been

constructed

Now

the

task

at

stage

i

can

be

considered

in

the

following

way

Add

all

links

of

weight

wi

to

the

forest

and

break

the

cycles

by

removing

links

of

weight

wi

Any

thinning

will

result

in

a

forest

of

partial

spanning

trees

of

maximal

weight

Note

that

any

thinning

at

a

given

stage

will

result

in

the

same

connected

components

and

therefore

the

thinning

chosen

has

no

impact

on

the

next

stage

Hence

if

we

in

the

construction

have

a

secondary

priority

-LRB-

distance

say

-RRB-

we

can

perform

the

thinning

by

using

Kruskal

's

algorithm

according

to

distance

In

this

way

we

will

end

up

with

a

maximal

weight

spanning

tree

of

maximal

distance

-LRB-

see

Figure

-RRB-

We

conclude

these

considerations

with

Theorem

A

maximal

distance

junction

tree

can

be

constructed

by

successively

choosing

a

link

of

maximal

weight

not

introducing

cycles

and

if

several

links

may

be

chosen

then

a

link

of

maximal

distance

is

selected

A

proof

of

Theorem

is

an

induction

proof

over

the

stages

The

induction

hypothesis

is

that

at

the

end

of

each

stage

the

forest

consists

of

partial

maximal

distance

junction

trees

Remark

An

analoguous

algorithm

based

on

Prim

's

algorithm

will

also

construct

maximal

distance

junction

trees

i

i

i

i

i

Figure

The

thinning

task

at

stage

i

in

Kruskal

's

algorithm

Corollary

All

junction

trees

over

the

same

triangulated

graph

have

the

same

separators

-LRB-

also

counting

multiplicity

-RRB-

Proof

Consider

stage

i

-LRB-

Figure

-RRB-

A

cycle

can

be

broken

by

removing

any

link

of

weight

wi

If

-LRB-

U

V

-RRB-

with

separator

S

is

removed

then

all

separators

in

the

remaining

paths

between

U

and

V

must

contain

S

This

means

that

any

separator

of

weight

wi

on

these

paths

must

equal

S.

By

thinning

we

therefore

remove

the

same

separators

Almond

trees

Almond

and

Kong

-LRB-

-RRB-

suggest

another

type

of

junction

tree

Compared

to

the

junction

trees

in

-LRB-

Jensen

et

al.

-RRB-

they

give

some

re

ABCABABDAAHAEFAEAEGAIABCABABDAAHAEFAEAEGAI

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

-LRB-

a

-RRB-

Contraction

of

the

junction

tree

in

Figure

-LRB-

b

-RRB-

An

Almond

tree

duction

in

computational

complexity

but

the

propagation

algorithm

becomes

much

less

transparent

Observation

If

n

links

have

the

same

separator

the

communication

scheme

can

be

contracted

-LRB-

see

Figure

-RRB-

In

junction

trees

each

separator

holds

exactly

one

potential

table

where

the

marginal

last

communicated

is

stored

In

contracted

junction

trees

a

separator

with

n

neighbours

must

hold

at

least

n

potential

tables

to

store

marginals

communicated

from

neighbours

This

means

that

there

is

no

saving

in

space

There

is

however

a

saving

in

time

since

a

number

of

marginalizations

are

avoided

Observation

If

a

separator

is

a

subset

of

another

separator

they

can

be

linked

-LRB-

see

Figure

-RRB-

The

type

of

calculations

are

the

same

for

links

between

separators

as

for

links

between

separators

and

cliques

Due

to

the

corollary

we

know

for

each

separator

S

the

number

of

supersets

to

which

it

shall

be

linked

and

for

each

link

-LRB-

S

S

-RRB-

we

can

associate

a

local

cost

C

-LRB-

S

S

-RRB-

Junction

trees

simplified

through

these

two

observations

we

call

Almond

trees

The

construction

of

an

Almond

tree

may

go

as

follows

ffl

From

the

triangulated

graph

the

set

of

cliques

and

the

set

of

separators

-LRB-

including

multiplicity

-RRB-

is

established

This

can

be

done

through

elimination

in

the

triangulated

graph

but

it

is

not

important

for

our

considerations

ffl

For

each

separator

establish

links

to

all

cliques

and

separators

containing

it

ffl

For

each

separator

-LRB-

with

multiplicity

n

-RRB-

choose

n

links

to

supersets

without

introducing

cycles

Theorem

An

optimal

Almond

tree

can

be

constructed

by

successively

choosing

links

for

separators

of

maximal

weight

and

if

several

links

may

be

chosen

take

one

of

minimal

complexity

A

proof

of

Theorem

is

an

induction

proof

along

the

same

line

as

a

proof

of

Theorem

References

Almond

R.

and

Kong

A.

-LRB-

-RRB-

Optimality

issues

in

constructing

a

Markov

tree

from

graphical

models

Journal

of

Computational

and

Graphical

Statistics

Jensen

F.

V.

-LRB-

-RRB-

Junction

trees

and

decomposable

hypergraphs

Research

report

Judex

Datasystemer

Aalborg

Denmark

Jensen

F.

V.

Lauritzen

S.

L.

and

Olesen

K.

G.

-LRB-

-RRB-

Bayesian

updating

in

causal

probabilistic

networks

by

local

computations

Computational

Statistics

Quarterly

-LCB-

Lauritzen

S.

L.

and

Spiegelhalter

D.

J.

-LRB-

-RRB-

Local

computations

with

probabilities

on

graphical

structures

and

their

application

to

expert

systems

Journal

of

the

Royal

Statistical

Society

Series

B

-LRB-

Methodological

-RRB-

-LRB-

-RRB-

-LCB-

Shachter

R.

D.

Andersen

S.

K.

and

Szolovits

P.

-LRB-

-RRB-

The

equivalence

of

exact

methods

for

probabilistic

inference

on

belief

networks

Technical

report

Department

of

Engineering-Economic

Systems

Stanford

University

Shibata

Y.

-LRB-

-RRB-

On

the

tree

representation

of

chordal

graphs

Journal

of

Graph

Theory

-LRB-

-RRB-

-LCB-

Zhang

N.

L.

and

Poole

D.

-LRB-

-RRB-

Sidestepping

the

triangulation

problem

in

Bayesian

net

computations

In

Dubois

D.

Wellman

M.

P.

D'Ambrosio

B.

and

Smets

P.

editors

Proceedings

of

the

Eighth

Conference

on

Uncertainty

in

Artificial

Intelligence

pages

-LCB-

Stanford

California

Morgan

Kaufmann

San

Mateo

California

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O 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bayesian networks
junction tree
markov networks
triangulation