directed mathematics computer science directed finite directed directed cycles consists edges edge directed way start follow consistently directed sequence edges eventually loops directed ordering sequence edge directed earlier later sequence model different kinds information relation forms partial order finite partial order represented program evaluation review technique uses model activities large human projects schedule projects use little total time possible scheduling computer operations updates compilation operations performed logic blocks electronic circuit design operations programming languages involve networks processing elements represent collections events influence structure network record historical data family trees version histories distributed revision control systems compact representation sequence data directed word representation collection strings decision representation sequences choices important time problems include finding ordering construction closure reduction largest smallest relation respectively closure problem goal find minimum weight edges connecting rest transforming directed cycles edges possible feedback set feedback edge set problem respectively hard directed contracting strongly connected component single problems finding shortest paths longest paths solved linear time contrast arbitrary shortest path slower longest path problems hard corresponding concept forest cycles choosing orientation forest produces special kind directed called kinds directed formed edges orientation assignment direction edges makes directed emphasize thing directed versions authors directed definitions formed collection edges objects connected pairs edges case directed edge orientation path directed described sequence edges having property ending edge sequence starting edge sequence path forms cycle starting edge equals ending edge directed directed cycles directed said exists path starts ends special case considered path zero edges reach path path edges path cycle way define directed reach path ordering directed ordering sequence edge start edge occurs earlier sequence ending edge ordering cycles edge earliest cycle oriented wrong way ordering conversely directed ordering property alternative definition directed exactly mathematical properties closure reduction relationship directed partial order partial order ordered exactly exists directed path different rise relation partial order example edges relation edges produce partial order ordered closure edges represents relation edge reach edge related pair distinct elements relation thought direct translation relation terms method translating partial orders works generally finite partially ordered set member edge pair elements related automatically closed relation way finite partially ordered set represented relation reduction fewest edges represents relation formed edges contains longer path connecting like closure reduction uniquely defined contrast directed minimal relation relation described partial order reduction edge pair covering relation reductions useful partial orders represent fewer edges representing orders lead simpler drawings partial order drawing reduction orientation edge shown placing starting edge lower position ending ordering directed ordering ordering starting edge occurs earlier ordering ending edge existence ordering characterize directed ordering general ordering unique unique ordering directed path containing case ordering order appear path family family linear extensions relation representing partial order set orders problem counting directed studied robinson number labeled restrictions order numbers appear ordering sequence numbers relation formula proved numbers count positive real numbers proof matrix matrix matrix positive denotes identity matrix self loops matrix zero diagonal adding preserves property matrix related families directed formed edges free tree particular true formed directing edges roots tree called strongly directed directed path direction forms tree problems recognition problem finding ordering given solved linear time builds ordering directly maintains list incoming edges included partially constructed ordering initially list consists incoming edges repeatedly adds list end partially constructed ordering checks neighbors added list terminates processed way alternatively ordering constructed reversing numbering depth search possible check given directed linear time attempting find ordering testing edge resulting ordering valid alternatively successfully orders meeting error condition construction choosing total order directing edge earlier order later resulting orientation edges called orientation different total orders lead orientation fewer number equal given directed removing feedback set feedback arc set set edges respectively touches cycles smallest set hard find arbitrary directed transformed called contracting strongly connected components single smallest feedback sets feedback arc sets closure reduction closure given edges constructed time breadth search depth search test alternatively solved time fast matrix theoretical improvement bound dense closure possible distinguish pairs path length pairs connected length path reduction consists edges form length paths paths connecting reduction constructed time bounds closure closure problem closure problem takes input directed weights seeks minimum maximum weight closure set outgoing edges problem formulated directed assumption greater case equivalent problem solved time reduction maximum flow problem path simpler instead general based principle ordering example possible find shortest paths longest paths given starting linear time processing order calculating path length minimum maximum length obtained incoming edges contrast arbitrary shortest path require slower ford longest paths arbitrary hard find applications scheduling directed representations partial applications scheduling systems tasks ordering constraints important class problems type concern collections objects need updated cells cells changed object files piece computer software source code changed context object updated edge connecting objects needs updated earlier cycle called circular generally allowed way consistently schedule tasks involved cycle circular form instance cell changes necessary values cells depend directly indirectly changed cell problem tasks scheduled values individual cells arise expression cell uses value cell case value earlier expression uses ordering order schedule cell updates allows updated single evaluation cell similar problems task ordering arise program compilation instruction scheduling low level computer program somewhat different based scheduling constraints program evaluation review technique method management large human projects applications method represent project specific tasks performed instead task activity represented edge connecting mark beginning completion task edge labeled estimate time team workers perform task longest path represents critical path project controls total time project individual scheduled according lengths longest paths ending data processing networks directed represent network processing elements representation data enters processing element incoming edges leaves element outgoing edges instance electronic circuit design static logic blocks represented system logic gates function input input output function represented individual bits general output blocks input captured register state element maintains properties electronic circuit paper database form directed instances components form directed reference lower level component electronic circuits necessarily directed programming languages describe systems operations data streams connections operations languages convenient describing repetitive data processing tasks connected collection operations applied data items executed parallel operation performed parallel process soon set available straight line code sequences statements loops branches represented describing operations performed code representation allows perform common elimination efficiently structures representing events edges representing relations events arranging linear order time arrows point direction time parent child affecting future past loops instance network represents system events directed likelihood event calculated predecessors context moral created adding edge parents called marrying replacing directed edges edges type similar structure influence represent decisions unknown information edges represent influences instance estimate expected value different choices intervention version history family trees seen directed family member edge parent child relationship despite necessarily trees possibility marriages relatives child common ancestor mother father causing pedigree collapse descent mother relationships women descent father relationships men trees ancestor family trees reason version history distributed revision control system generally structure directed revision edge connecting pairs revisions directly derived trees general merges geometry maintains history representing version history geometric structure course sequence changes structure instance changes replacing triangle smaller point added flip operations replace pairs different pair history triangle constructed edges triangle replace structure allows point location answered efficiently find location point follow path history step moving replacement triangle contains final triangle reached path triangle contains data compression directed compact representation collection sequences type application finds paths form given sequences sequences share shared represented shared allowing representation use space list sequences separately example directed word data structure computer science formed directed single source edges labeled letters symbols paths source represent set strings english words set sequences represented paths tree forming tree prefix sequence making parent represent sequence fewer element tree formed way set strings called directed word saves space allowing paths rejoin set words possible represented single tree idea represent family paths occurs decision based data structure representing functions decision non sink labeled variable sink edge labeled function value truth assignment value sink found following path starting single source non sink follows outgoing edge labeled value variable directed word viewed compressed form tries decision viewed compressed forms decision trees save space allowing paths rejoin agree results remaining decisions
Directed acyclic graph