Algorithms and Data Structures

Transcription

1 Algorithms and Data Structures PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Summer Term 08

2 Part : Graphs The four colour problem dating back to 8 source: wikipedia.org source: math.gatech.edu PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

3 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

4 Fundamentals definitions relation U V fundamental formalism in order to express relations between entities of basic sets U and V example U : set of airlines LH, LX, OS, BA, EN, AF, IB,... V : set of airports MUC, HAM, LHR, ZRH, DUB, DBX, EWR,... : relation airline u flies to v with u U, v V Cartesian product U V set of all ordered pairs (u, v) with u U, v V LH BA LX EN U U V V SEA ZRH BHX MUC FLR LH BA LX EN U U V V SEA ZRH BHX MUC FLR PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

5 Fundamentals definitions (cont d) usual case: homogeneous relation V V with (finite) basic set V set of persons V with relation person x knows person y with x, y V given: basic set V v i of elements v i that are numbered with i 0,..., n and which is called set of vertices hence, a directed graph G (V, E) is given by the set of vertices V and some relation E V V elements (v, v ) E are called edges and E as the set of edges; instead of (v, v ) more often edges are written as v v example a b a b a b c d c d c d different representations of the same directed graph PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

6 Fundamentals definitions (cont d) if relation is symmetric, then with v v also v vbelongs to E the relation between vertices v, v is said to be mutual graphs with these properties are called general or undirected graphically, edges are not drawn as arrows but as simple lines a b c d source: mvg.de network: undirected graph with V as set of stations and E as connexion between always two stations of commuter trains PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 6

7 Fundamentals definitions (cont d) the inverse relation conforms to the dual graph G T (V, E T ) with (v, v ) E T if and only if (v, v) E hence, in G T all edges are reversed, i.e. from v v follows v v obviously, if holds the graph is undirected the output set v of some vertex v is defined as set of edges starting in v correspondingly, the input set v is defined as set of edges ending up in v v and v are called in degree or out degree, resp., of some vertex v summing up over all vertices of a directed graph the following holds in undirected graphs, every edge starting in some vertex v also ends up in the same vertex v, hence v v for all v V more general, one speaks of the degree deg(v) of some vertex v PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 7

8 Fundamentals something more about graphs a sequence (v 0, v,..., v n ) of edges (v i, v i ) E with i 0,..., n is called path of length n from vertex v 0 to vertex v n v n is reachable from v 0 a path of length n is called cycle if v 0 v n a directed graph (DG) that contains no directed cycles (i.e. with all edges being oriented in the same direction) is called acyclic v 0 v v 0 v 3 path of length n cycle of length n 3 a cycle is called simple if all vertices v i with i n are different it is called Euler cycle if each edge of the DG is visited exactly once it is called Hamiltonian cycle if each vertex of the DG is visited exactly once PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 8

9 Technische Universität München Fundamentals something more about graphs (cont d) example: Seven Bridges of Königsberg Königsberg set on both sides of Pregel River with islands A (dome), B and mainland portions C, D all connected by seven bridges find (closed) walk through city crossing each bridge only once C deg(c) 3 C deg(b) 3 deg(a) A source: landkartenarchiv.de A B B deg(d) 3 D D solution given by L. EULER (763): closed walk if and only if each vertex has an even degree until 9 capital of East Prussia and residence of Kant, Kollwitz, E.T.A. Hoffmann, Hilbert,...; since 96 Kaliningrad (today Russia) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 9

10 Fundamentals something more about graphs (cont d) path (v, v) of length n is called loop, i.e. connecting a vertex to itself any relation is called reflexive iff for each vertex exists one loop is called irreflexive if no cycle of length n exists ( simple graph) an undirected graph is called complete if every pair of distinct vertices is connected by a unique edge also referred to as graph E n with n V denoting the number of vertices complete graphs E 3, E, E, and E 6 question: how to store graphs efficiently...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 0

11 Fundamentals storage: adjacency matrix let G (V, E) be a directed graph with vertices v i numbered from,..., n storage of G as adjacency matrix, i.e. as n n matrix A (a ij ) with a ij, if (v i, v j ) K 0, otherwise for set E V V example v v v 3 v v v v 0 A v v 3 v 3 v 0 0 v for undirected graphs follows with a ij also a ji, hence A A T, i.e. A is symmetric PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

12 Fundamentals storage: adjacency list let G (V, E) be a directed graph with vertices v i numbered from,..., n if G is not dense, storage as adjacency list is advantageous idea: for each vertex all adjacent vertices are stored within a linked list example v v v v v 3 v v v 3 v 3 v 3 v v v v 3 for undirected graphs follows with (v i, v j ) E that list v i has to store element [v j ] and list v j has to store element [v i ] redundancy simple operations such as deletion of a vertex are not properly supported by this representation list of adjacent vertex has to be modified, too PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

13 Fundamentals storage: incidence matrix let G (V, E) be an irreflexive graph with vertices v i numbered from,..., n and edges e j numbered from,..., m storage of G as incidence matrix, i.e. as n m matrix B (b ij ) with, if e j (v i, ) b ij 0, if v i e j, if e j (, v i ) or b ij, if v i e j 0, otherwise for directed graphs for undirected graphs for set E V V in case of directed graphs, each column of B contains exactly one (start vertex) and one (end vertex); in case of undirected graphs, each column of B has two entries different from 0 PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

14 Fundamentals storage: incidence matrix (cont d) example e e e 3 e e e 6 e v v v e e B v v 3 e 3 e v 3 v e v observation: each column of B sums up to zero résumé as always, representation is problem dependent typically, graphs are stored as adjacency matrix caution when storing huge graphs ( memory footprint (!)) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

15 Fundamentals edge weighted graphs let G (V, E) be a directed or undirected graph each edge e has one additional attribute w(e), called weight instead of (i.e. e : v i v j E) the adjacency matrix A (a ij ) stores the corresponding weight w(e); possible meanings length in [m] cost in [ ] flow in [l s] example v v v 3 v v 3 v v 0 3 A v v 3 v 3 v 0 0 v PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

16 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 6

17 Minimum Spanning Tree knocking on wood... let G (V, E) be an acyclic undirected graph with edge weights w(e) G is called tree if each two vertices are connected via one path only for trees V E holds as for one edge less the graph would not be connected for one edge more the graph would contain at least one cycle definition: spanning tree A set of edges T E is called spanning tree of G, if and only if (V, T) is a tree. graph G possible spanning trees of G PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 7

18 Minimum Spanning Tree knocking on wood... (cont d) weight of subset E Edefined as definition: minimal spanning tree A spanning tree T E is called minimal spanning tree of G, if there exists no spanning tree of G with smaller weight than w(t). for instance, spanning trees are an important topic within planning of communication networks or electric power grids reachability of all vertices (e.g. cities) with minimal cost (e.g. cables, excavations,...) computation of minimal spanning tree via algorithm from KRUSKAL (96) or PRIM (97) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 8

19 Minimum Spanning Tree PRIM algorithm let T be the set of edges selected so far forming a minimum spanning tree then further edges are chosen thus w(t) is increased by a minimum only hence, the next edge (v, v ) to be included in T ) has to be a minimum cost edge not in T ) with the property that T (v, v ) is also a tree T possible next edges edge violates rule ) the idea for this algorithm is referred to the American computer scientist R.C. PRIM, nevertheless it was originally discovered in 930 by the Czech mathematician V. JARNÍK and, thus, is also called PRIM JARNÍK algorithm PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 9

20 Minimum Spanning Tree PRIM algorithm (cont d) preliminary: set of edges T is empty and cost C(T) 0. find edge e : (v i, v j ) with minimal weight w(e). tag vertices v i, v j, store e in T, and update C(T) C(T) w(e) initialisation (V ) 3. find next edge e : (v, v ) such that loop over all vertices i. vertex v is tagged ii. vertex v is untagged iii. edge e has minimal weight w(e ) depends on the implementation. tag vertex v, store e in T, and update C(T) C(T) w(e ). repeat steps 3. and. until all vertices are tagged for simplicity, V (instead of V ) denotes the number or vertices and E (instead of E ) the number of edges PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 0

21 PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 Technische Universität München 6 7 E D E C D E 6 7 Minimum Spanning Tree PRIM algorithm (cont d) example C 3 6 B D E 7 7 D B E 6 7 E D C C(T) C(T) 3 C(T) E D C(T) E C(T) E B C D F A A F C(T) 0 F A A B F 3 A B C D E F A B C D F A B C F

22 Minimum Spanning Tree implementation and complexity how to find the next edge efficiently...? idea: associate with each vertex v a value NEAR(v) thus NEAR(v) 0 in case vertex v is already flagged (and included in T) NEAR(v) 0 and A(v, NEAR(v)) is minimum otherwise hence, next edge is defined by NEAR(v) 0 and A(v, NEAR(v)) is minimum note: in case (v, v ) E the regarding coefficient must be set to a ij initialisation of NEAR( ), assuming v, v are the first two vertices being flagged and, thus, e : (v, v ) being the edge with minimal weight w(e) of G for i 0 to V do if A(i, v) A(i, v ) then NEAR(i) v else NEAR(i) v fi od NEAR(v) 0; NEAR(v ) 0 (V) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

23 Minimum Spanning Tree implementation and complexity (cont d) algorithm fragment conforms to E V times for i to V do find next edge with NEAR(v) 0 and A(v, NEAR(v)) minimum update... for k 0 to V do if NEAR(k) 0 and A(k, v) A(k, NEAR(k)) then NEAR(k) v fi od od (V) complexity start: finding minimal edge and initialisation of NEAR( ) (V ) algorithm: V times update of NEAR( ) (V ) hence, PRIM belongs to class (V ) with quadratic runtime complexity optimised versions for sparse graphs with ((E V) log (V)) steps PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

24 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

25 Single Source Shortest Path routing problem: what is the shortest path from x to y... turn left in 0 metres take first exit left drive 900 metres and arrive at destination source: maps.google.de 7 A E F 3 B D 6 C let G (V, E) be a simple graph with positive edge weights w(e) G stored as adjacency matrix with a ij in case (v i, v j ) E PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

26 Single Source Shortest Path routing (cont d) let v 0 and v k be two arbitrary vertices of graph G i. is there a path from v 0 to v k...? ii. if there is more than one path, which is the shortest...? both i. and ii. are special cases of the general path problem single source shortest path (SSSP): starting from one dedicated vertex s, find all shortest paths to all other vertices v V SSSP was studied by E.W. DIJKSTRA and the regarding algorithm is referred to as DIJKSTRA algorithm (99) GO TO considered harmful. (960) source: thocp.net PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 6

27 Single Source Shortest Path something about complexity... combinatorial explosion (brute force search) s e... e layer layer 0 number of vertices V #layers V 0 number of paths P #layers P 0,,899,906,8,6 assumption: computer is able to test,000,000 path every second runtime for above problem: 36 years (!) with 60 layers 37,000 years; with 70 layers 37,000,000 years PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 7

28 Single Source Shortest Path something about complexity... (cont d) idea: do not test all paths, only investigate further in those which have been found shortest so far w(e) denotes the length of edge e : v v, also written as w(v, v ) length of path p (v 0, v,..., v k ) from v 0 ( s) to v k ( t) is defined as definition: shortest path p is called shortest path from s to t if there is no path p from s to t with w(p ) w(p). The length of path from s to t is called distance and denoted with (s, t). In case there exists no path from s to t then (s, t). solution to above problem DIJKSTRA algorithm PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 8

29 Single Source Shortest Path DIJKSTRA algorithm compute spanning tree which starting in some vertex s V successively finds all shortest paths to all other vertices thus, shortest paths (SP) are computed in nondecreasing order, i.e. find path to nearest vertex first, then to second nearest, then to third,... Let p (v 0, v,..., v k ) be the shortest path from v 0 to v k. For each pair i, j with 0 i j k path p ij (v i, v i,..., v j ) is then a shortest path from v i to v j. assume, path from v 0 to v k contains vertex u to which no SP was found yet then path looks like (v 0,..., u,..., v k ) with w(v 0, u) w(v 0, v k ) this violates our assumption paths are generated in nondecreasing order, hence a shortest path to u must already have been found furthermore: for shortest path p (v 0, v,..., v k ) from v 0 to v k holds (v 0, v k ) (v 0, v k ) w(v k, v k ) important for finding next edge to be included in T PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 9

30 Single Source Shortest Path DIJKSTRA algorithm (cont d) preliminary: set of edges T is empty and start vertex s has been chosen. initialise distance to each vertex v V with (s, v). tag vertex s, set (s, s) 0, and continue with step. (where v s) initialisation 3. find next vertex v loop over all vertices i. that is not tagged ii. whose distance (s, v ) is minimal. tag vertex v and store edge e : (v p, v ) from predecessor v p to v in T. test for all untagged vertices v if a shorter path via v exists, thus that (s, v ) w(v, v ) (s, v ) and if so update (s, v ) and store v as predecessor of v 6. repeat steps 3. to. until all vertices are tagged PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 30

31 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E A 7 E F B 6 D C A B C D E F v p v p v p v p v p 0 B B 6 B B v p PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

32 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E A B C D E F v p v p v p v p v p v p A B 0 B B 6 B B F C B 6 B 9 A 3 A 7 E 6 D PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

33 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E 7 A E F B 6 D C A v p B B v p 0 C v p D v p E v p F v p B 6 B B B 6 B 9 A 3 A B 6 B 8 F PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 33

34 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E 7 A E F B 6 D C A v p B B v p 0 C v p D v p E v p F v p B 6 B B B 6 B 9 A 3 A B 6 B 8 F C 8 F PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

35 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E 7 A E F B 6 D C A v p B B v p 0 C v p D v p E v p F v p B 6 B B B 6 B 9 A 3 A B 6 B 8 F C 8 F 7 D PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

36 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E 7 A E F B D 6 C A v p B B v p 0 C v p D v p E v p F v p B 6 B B B 6 B 9 A 3 A B 6 B 8 F C 8 F 7 D PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 36

37 Single Source Shortest Path DIJKSTRA algorithm (cont d) example: starting point is vertex B with destination E A E F B D C A B C D E F v p v p v p v p v p v p 0 B B 6 B B B 6 B 9 A 3 A B 6 B 8 F C 8 F 7 D B 0 B C 7 D 3 A path: B C D E with (B, E) 7 algorithm finds all shortest paths from vertex B to all other vertices for different starting points typically different spanning trees are computed PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 37

38 Single Source Shortest Path complexity initialisation set all distances (s, v) to infinity (step.) (V) update all distances (s, v) routing through s (step.) (V) loop over all vertices find next vertex v (step 3.) (V) update all distances (s, v ) routing through v (step.) (V) repeat above steps V times total cost (V ) hence, DIJKSTRA belongs to class (V ) with quadratic runtime complexity optimised versions use a heap for choosing the minimum and, thus, have a complexity of ((E V) log (V)) variants: A ( informed DIJKSTRA using heuristics), BELLMANN FORD (allows negative edge weights), FLOYD WARSHALL (all pairs shortest path (APSP)) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 38

39 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 39

40 Transitive Closure fundamentals definition: some relation R V V is called transitive if for all vertices v, v, v Vwith v v and v v follows v v example: from 3 and 3 due to transitivity follows with R we denote the transitive closure of R that contains all pairs (v, v ) that are connected through R, i.e. there is (at least) one path from v to v in other words: R contains all paths in a directed graph, i.e. with v v and v v in R follows that v v is also included in R example v 3 v v 3 v 0 R R v v v v PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 0

41 Transitive Closure fundamentals (cont d) idea: a path exists between two vertices v i and v j if there is an edge from v i to v j, or a path from v i to v j through vertex v, or a path from v i to v j through vertex v and or v, or a path from v i to v j through vertex v, v and or v 3, or... more formal v i v v v 3 v j v i v k (v i, v j ) R(k) (v i, v j ) R (k ) (v i, v k ) R (k ) and (v k, v j ) R (k ) v j during k th iteration, the algorithm determines if a path between two vertices v i and v j exists using just vertices v m with m,..., k allowed as an intermediate PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

42 Transitive Closure WARSHALL algorithm previous recurrence (v i, v j ) R (k) (v i, v j ) R (k ) ((v i, v k ) R (k ) (v k, v j ) R (k ) ) leads to the following rules ) if an element r ij in R (k ), it remains in R (k) ) if an element r ij 0 in R (k ), it has to be changed to in R (k) if and only if both elements r ik and r kj in R (k ) j k j k R (k ) k R (k) k 0 i i PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

43 Transitive Closure WARSHALL algorithm (cont d) preliminary: directed graph G (V, E) stored as adjacency matrix A R (0) A for k to V do for i to V do for j to V do r(i, j) (k) r(i, j) (k ) (r(i, k) (k ) r(k, j) (k ) ) od od od what about complexity...? there are three nested loops over i, j, k obviously, WARSHALL belongs to class (V 3 ) and has a cubic runtime complexity, making it very slow for large adjacency matrices PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

44 Transitive Closure WARSHALL algorithm (cont d) example a b R (0) a b c d a b c a b R () a b c d a b c c d 0 0 d c d 0 0 d a b R () a b c d a b c a b R () a b c d a b c c d 0 d c d d R PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

45 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

46 Network Flow fundamentals let G (V, E) be a directed graph with two designated vertices s, t V, called source and sink, resp., as well as a capacity function c : V V, assigning each edge (v, v ) E a non negative capacity c(v, v ) we further assume, each vertex is reachable from source s network flow is a function f : V V that fulfils following properties. conservation of flow: for all vertices v V \ s, t holds. limitation of capacity: for all vertices v, v Vholds 0 f(v, v ) c(v, v ) hence, we define the total flow F : V V as or PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 6

47 Network Flow fundamentals (cont d) application: many questions can be modelled as flow problem assume, ports A,..., A p store bananas for shipping to ports B,..., B q r i with i,..., p denotes how many bananas are ready for shipping at A i d j with j,..., q denotes how many bananas are requested at port B j c(a i, B j ) with i,..., p and j,..., q denotes the capacity of bananas (in metric tons) that can be transported via route between ports A i and B j question. is it possible to satisfy all request. if not, how many tons can be transported at most to all destinations 3. how (i.e. on which route) should the bananas be shipped all questions above to be modelled as flow problem using a directed graph G (V, E) with V s, t, A,..., A p, B,..., B q PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 7

48 Network Flow fundamentals (cont d) sample graph for p 3 and q s r r A A c(a, B ) B d t let F : E A 3 be a total flow, thus previous question easily to be answered. all request can be satisfied if F. at most F tons of bananas can be transported to all destinations 3. flow f(a i, B j ) denotes how many tons are transported from A i to B j other flow problem example: can a soccer team (still) become champion according to the current position in the league and the remaining matches B d r 3 c(a 3, B ) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 8

49 Network Flow fundamentals (cont d) problem: how to find a flow f for which the total flow is maximum a partitioning of V into parts S V and T V is called cut with s S, t T, and T V \ S; let c(s, T) denote the capacity of the cut (S, T) defined as the flow f over a cut (S, T) is then defined as example s 3 T t obviously, capacity c(s, T) is an upper bound for the flow that can be sent from source to sink S c(s, T) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 9

50 Network Flow fundamentals (cont d) from previous observation, we can reason that the total flow can be measured at any arbitrary cut (S, T) and is bounded by c(s, T) F f(s, T) c(s, T) proof: with 0 f(e) c(e) for every edge e E follows () hence, if F and c(s, T) satisfy () by equality then F is maximum and the cut (S, T) is of minimum capacity remark: if F is maximum the flow cannot be further increased any algorithm for updating flows f(e) must halt termination is guaranteed PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 0

51 Network Flow FORD and FULKERSON algorithm task: compute for G (V, E) with capacities c : V V its total flow idea: find augmenting path from source to sink such that the flow along this path can be increased without exceeding capacity limitations algorithm proposed by L.R. FORD, JR. and D.R. FULKERSON in 96. initialise flow f(e) 0 for all e E. while augmenting path p (s v 0, v, v,..., v k t) i. increase flow f along p 3. compute total flow F definition: augmenting path The residual network G f (V, E f ) is defined on the same set of vertices V and on edges (v, v ) E f where (v, v ) E c f (v, v ) c(v, v ) f(v, v ) 0 or (v, v) E c f (v, v ) f(v, v) 0 holds. An augmenting path is a simple path from s to t in G f. PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

52 Network Flow FORD and FULKERSON algorithm (cont d) example 6 3 s t s t left: G with edge labelling a b denoting a f(e) and b c(e); right: residual network G f s 6 t s t two possible augmenting paths PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

53 Network Flow FORD and FULKERSON algorithm (cont d) finally, what means increasing flow along some augmenting path...? let p be a simple path from s to t within the residual network G f then, the flow along that path is increased f(v, v ) f(v, v ) f(v, v ) if (v, v ) E (v, v ) p if (v, v ) E (v, v) p f(v, v ) otherwise with note: if all capacities of G are integer values, then the total flow also is an integer value ( important property for some applications) PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 3

54 Network Flow FORD and FULKERSON algorithm (cont d) example: full processing of FFA (w/o explicit presentation of G f ) s v v t s v v t 0 7 v 3 v initial situation all flows f(e) v 3 v path s v v v 3 v t with s 6 v v t s 0 6 v 6 0 v t v v 3 path s v 3 v t with v v 3 path s v 3 v t with PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

55 Network Flow FORD and FULKERSON algorithm (cont d) example: full processing of FFA (w/o explicit presentation of G f ) s v v 0 v 3 v path s v v t with t complexity for any given flow residual network G f can be computed in (E) time any path from s to t in G f can be found in (E) time for integer capacities, the flow is increased at least by one during each iteration, thus the number of iterations is bounded by C hence, FFA has (E C) runtime complexity which grows for increasing E s v v 0 T S v 3 v no more augmenting path to be found total flow F t PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08

56 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 6

57 What Else...? travelling salesperson problem (TSP) formerly known as travelling salesman problem question: given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city Hamiltonian cycle source: Jessica Yu solution to 8 states travelling salesperson problem TSP is a combinatorial optimisation problem no algorithm can find the solution in polynomial worst case runtime PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 7

58 overview fundamentals minimum spanning tree (single source) shortest path transitive closure network flow what else...? PD Dr. Ralf Peter Mundani Algorithms and Data Structures Summer Term 08 8