CS 5114: Theory of Algorithms. Graph Algorithms. A Tree Proof. Graph Traversals. Clifford A. Shaffer. Spring 2014

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1 epartment of omputer Science Virginia Tech lacksburg, Virginia opyright c 04 by lifford. Shaffer : Theory of lgorithms Title page : Theory of lgorithms lifford. Shaffer Spring 04 lifford. Shaffer epartment of omputer Science Virginia Tech lacksburg, Virginia Students should be familiar with inductive proofs, recursion, data structures, and programming at the S4 level. Spring 04 opyright c 04 by lifford. Shaffer : Theory of lgorithms Spring 04 / 60 Graph lgorithms Graph lgorithms Graph lgorithms Graphs are useful for representing a variety of concepts: ata Structures Relationships amilies ommunication Networks Road Maps Graphs are useful for representing a variety of concepts: ata Structures Relationships amilies ommunication Networks Road Maps : Theory of lgorithms Spring 04 / 60 Tree Proof efinition: free tree is a connected, undirected graph that has no cycles. Theorem: If T is a free tree having n vertices, then T has exactly n edges. Proof: y induction on n. ase ase: n =. T consists of vertex and 0 edges. Inductive Hypothesis: The theorem is true for a tree having n vertices. Inductive Step: If T has n vertices, then T contains a vertex of degree. Remove that vertex and its incident edge to obtain T, a free tree with n vertices. y IH, T has n edges. Thus, T has n edges. graph G = (V, ) consists of a set of vertices V, and a set of edges, such that each edge in is a connection between a pair of vertices in V. irected vs. Undirected Labeled graph, weighted graph Labels for edges vs. weights for edges Multiple edges, loops ycle, ircuit, path, simple path, tours ipartite, acyclic, connected Rooted tree, unrooted tree, free tree Tree Proof Tree Proof efinition: free tree is a connected, undirected graph that has no cycles. Theorem: If T is a free tree having n vertices, then T has exactly n edges. Proof: y induction on n. ase ase: n =. T consists of vertex and 0 edges. Inductive Hypothesis: The theorem is true for a tree having n vertices. Inductive Step: If T has n vertices, then T contains a vertex of degree. Remove that vertex and its incident edge to obtain T, a free tree with n vertices. y IH, T has n edges. Thus, T has n edges. This is close to a satisfactory definition for free tree. There are several equivalent definitions for free trees, with similar proofs to relate them. Why do we know that some vertex has degree? ecause the definition says that the ree Tree has no cycles. : Theory of lgorithms Spring 04 / 60 Graph Traversals Graph Traversals Graph Traversals Various problems require a way to traverse a graph that is, visit each vertex and edge in a systematic way. Three common traversals: ulerian tours Traverse each edge exactly once epth-first search Keeps vertices on a stack readth-first search Keeps vertices on a queue Various problems require a way to traverse a graph that is, visit each vertex and edge in a systematic way. a vertex may be visited multiple times Three common traversals: ulerian tours Traverse each edge exactly once epth-first search Keeps vertices on a stack readth-first search Keeps vertices on a queue : Theory of lgorithms Spring 04 4 / 60

2 (a) (b) ulerian Tours ulerian Tours ulerian Tours circuit that contains every edge exactly once. xample: f e a c b d Tour: b a f c d e. f xample: e a c b d g No ulerian tour. How can you tell for sure? circuit that contains every edge exactly once. xample: f e a c b d Tour: b a f c d e. f xample: e a c b d g No ulerian tour. How can you tell for sure? Why no tour? ecause some vertices have odd degree. ll even nodes is a necessary condition. Is it sufficient? : Theory of lgorithms Spring 04 / 60 ulerian Tour Proof Theorem: connected, undirected graph with m edges that has no vertices of odd degree has an ulerian tour. Proof: y induction on m. ase ase: Inductive Hypothesis: Inductive Step: Start with an arbitrary vertex and follow a path until you return to the vertex. Remove this circuit. What remains are connected components G, G,..., G k each with nodes of even degree and < m edges. y IH, each connected component has an ulerian tour. ombine the tours to get a tour of the entire graph. ulerian Tour Proof ulerian Tour Proof Theorem: connected, undirected graph with m edges that has no vertices of odd degree has an ulerian tour. Proof: y induction on m. ase ase: Inductive Hypothesis: Inductive Step: Start with an arbitrary vertex and follow a path until you return to the vertex. Remove this circuit. What remains are connected components G, G,..., Gk each with nodes of even degree and < m edges. y IH, each connected component has an ulerian tour. ombine the tours to get a tour of the entire graph. ase case: 0 edges and vertex fits the theorem. IH: The theorem is true for < m edges. lways possible to find a circuit starting at any arbitrary vertex, since each vertex has even degree. : Theory of lgorithms Spring 04 6 / 60 epth irst Search epth irst Search epth irst Search void S(Graph G, int v) { // epth first search PreVisit(G, v); // Take appropriate action G.setMark(v, VISIT); for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) S(G, G.v(w)); PostVisit(G, v); // Take appropriate action Initial call: S(G, r) where r is the root of the S. ost: Θ( V + ). void S(Graph G, int v) { // epth first search PreVisit(G, v); // Take appropriate action G.setMark(v, VISIT); for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) S(G, G.v(w)); PostVisit(G, v); // Take appropriate action Initial call: S(G, r) where r is the root of the S. ost: Θ( V + ). : Theory of lgorithms Spring 04 7 / 60 epth irst Search xample epth irst Search xample epth irst Search xample The directions are imposed by the traversal. This is the epth irst Search Tree. (a) (b) : Theory of lgorithms Spring 04 8 / 60

3 S Tree If we number the vertices in the order that they are marked, we get S numbers. Lemma 7.: very edge e is either in the S tree T, or connects two vertices of G, one of which is an ancestor of the other in T. S Tree Results: No cross edges. That is, no edges connecting vertices sideways in the tree. S Tree If we number the vertices in the order that they are marked, we get S numbers. Lemma 7.: very edge e is either in the S tree T, or connects two vertices of G, one of which is an ancestor of the other in T. Proof: onsider the first time an edge (v, w) is examined, with v the current vertex. If w is unmarked, then (v, w) is in T. If w is marked, then w has a smaller S number than v N (v, w) is an unexamined edge of w. Thus, w is still on the stack. That is, w is on a path from v. Proof: onsider the first time an edge (v, w) is examined, with v the current vertex. If w is unmarked, then (v, w) is in T. If w is marked, then w has a smaller S number than v N (v, w) is an unexamined edge of w. Thus, w is still on the stack. That is, w is on a path from v. : Theory of lgorithms Spring 04 9 / 60 S for irected Graphs S for irected Graphs S for irected Graphs Main problem: connected graph may not give a single S tree orward edges: (, ) ack edges: (, ) 9 8 ross edges: (6, ), (8, 7), (9, ), (9, 8), (4, ) Solution: Maintain a list of unmarked vertices. Whenever one S tree is complete, choose an arbitrary unmarked vertex as the root for a new tree. Main problem: connected graph may not give a single S tree orward edges: (, ) ack edges: (, ) 9 8 ross edges: (6, ), (8, 7), (9, ), (9, 8), (4, ) Solution: Maintain a list of unmarked vertices. Whenever one S tree is complete, choose an arbitrary unmarked vertex as the root for a new tree. : Theory of lgorithms Spring 04 0 / 60 irected ycles Lemma 7.4: Let G be a directed graph. G has a directed cycle iff every S of G produces a back edge. irected ycles See earlier lemma. irected ycles Lemma 7.4: Let G be a directed graph. G has a directed cycle iff every S of G produces a back edge. Proof: Suppose a S produces a back edge (v, w). v and w are in the same S tree, w an ancestor of v. (v, w) and the path in the tree from w to v form a directed cycle. Suppose G has a directed cycle. o a S on G. Let w be the vertex of with smallest S number. Let (v, w) be the edge of coming into w. v is a descendant of w in a S tree. Therefore, (v, w) is a back edge. Proof: Suppose a S produces a back edge (v, w). v and w are in the same S tree, w an ancestor of v. (v, w) and the path in the tree from w to v form a directed cycle. Suppose G has a directed cycle. o a S on G. Let w be the vertex of with smallest S number. Let (v, w) be the edge of coming into w. v is a descendant of w in a S tree. Therefore, (v, w) is a back edge. : Theory of lgorithms Spring 04 / 60 readth irst Search readth irst Search readth irst Search Like S, but replace stack with a queue. Visit vertex s neighbors before going deeper in tree. Like S, but replace stack with a queue. Visit vertex s neighbors before going deeper in tree. : Theory of lgorithms Spring 04 / 60

4 J (a) J J J6 J4 J (b) J7 readth irst Search lgorithm void S(Graph G, int start) { Queue Q(G.n()); Q.enqueue(start); G.setMark(start, VISIT); while (!Q.ismpty()) { int v = Q.dequeue(); PreVisit(G, v); // Take appropriate action for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) { G.setMark(G.v(w), VISIT); Q.enqueue(G.v(w)); PostVisit(G, v); // Take appropriate action readth irst Search lgorithm readth irst Search lgorithm void S(Graph G, int start) { Queue Q(G.n()); Q.enqueue(start); G.setMark(start, VISIT); while (!Q.ismpty()) { int v = Q.dequeue(); PreVisit(G, v); // Take appropriate action for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) { G.setMark(G.v(w), VISIT); Q.enqueue(G.v(w)); PostVisit(G, v); // Take appropriate action : Theory of lgorithms Spring 04 / 60 readth irst Search xample readth irst Search xample readth irst Search xample Non-tree edges connect vertices at levels differing by 0 or. We know this because if an edge had connected to a deeper level, then that target node would have been placed on the queue when the edge was encountered. (a) (b) Non-tree edges connect vertices at levels differing by 0 or. : Theory of lgorithms Spring 04 4 / 60 Topological Sort Topological Sort Topological Sort Problem: Given a set of jobs, courses, etc. with prerequisite constraints, output the jobs in an order that does not violate any of the prerequisites. Problem: Given a set of jobs, courses, etc. with prerequisite constraints, output the jobs in an order that does not violate any of the prerequisites. J6 J J J J7 J J4 : Theory of lgorithms Spring 04 / 60 Topological Sort lgorithm void topsort(graph G) { // Top sort: recursive for (int i=0; i<g.n(); i++) // Initialize Mark G.setMark(i, UNVISIT); for (i=0; i<g.n(); i++) // Process vertices if (G.getMark(i) == UNVISIT) tophelp(g, i); // all helper void tophelp(graph G, int v) { // Helper function G.setMark(v, VISIT); for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) tophelp(g, G.v(w)); printout(v); // PostVisit for Vertex v Topological Sort lgorithm Prints in reverse order. Topological Sort lgorithm void topsort(graph G) { // Top sort: recursive for (int i=0; i<g.n(); i++) // Initialize Mark G.setMark(i, UNVISIT); for (i=0; i<g.n(); i++) // Process vertices if (G.getMark(i) == UNVISIT) tophelp(g, i); // all helper void tophelp(graph G, int v) { // Helper function G.setMark(v, VISIT); for (dge w = each neighbor of v) if (G.getMark(G.v(w)) == UNVISIT) tophelp(g, G.v(w)); printout(v); // PostVisit for Vertex v : Theory of lgorithms Spring 04 6 / 60

5 Queue-based Topological Sort void topsort(graph G) { // Top sort: Queue Queue Q(G.n()); int ount[g.n()]; for (int v=0; v<g.n(); v++) ount[v] = 0; for (v=0; v<g.n(); v++) // Process every edge for (dge w each neighbor of v) ount[g.v(w)]++; // dd to v s count for (v=0; v<g.n(); v++) // Initialize Queue if (ount[v] == 0) Q.enqueue(v); while (!Q.ismpty()) { // Process the vertices int v = Q.dequeue(); printout(v); // PreVisit for v for (dge w = each neighbor of v) { ount[g.v(w)]--; // One less prereq if (ount[g.v(w)]==0) Q.enqueue(G.v(w)); Queue-based Topological Sort Queue-based Topological Sort void topsort(graph G) { // Top sort: Queue Queue Q(G.n()); int ount[g.n()]; for (int v=0; v<g.n(); v++) ount[v] = 0; for (v=0; v<g.n(); v++) // Process every edge for (dge w each neighbor of v) ount[g.v(w)]++; // dd to v s count for (v=0; v<g.n(); v++) // Initialize Queue if (ount[v] == 0) Q.enqueue(v); while (!Q.ismpty()) { // Process the vertices int v = Q.dequeue(); printout(v); // PreVisit for v for (dge w = each neighbor of v) { ount[g.v(w)]--; // One less prereq if (ount[g.v(w)]==0) Q.enqueue(G.v(w)); : Theory of lgorithms Spring 04 7 / 60 Shortest Paths Problems Shortest Paths Problems Shortest Paths Problems Input: graph with weights or costs associated with each edge. Output: The list of edges forming the shortest path. Sample problems: ind the shortest path between two specified vertices. ind the shortest path from vertex S to all other vertices. ind the shortest path between all pairs of vertices. Input: graph with weights or costs associated with each edge. Our algorithms will actually calculate only distances. Output: The list of edges forming the shortest path. Sample problems: ind the shortest path between two specified vertices. ind the shortest path from vertex S to all other vertices. ind the shortest path between all pairs of vertices. Our algorithms will actually calculate only distances. : Theory of lgorithms Spring 04 8 / 60 Shortest Paths efinitions Shortest Paths efinitions Shortest Paths efinitions d(, ) is the shortest distance from vertex to. w(, ) is the weight of the edge connecting to. If there is no such edge, then w(, ) =. 0 0 d(, ) is the shortest distance from vertex to. w(, ) = 0; d(, ) = 0 (through ). w(, ) is the weight of the edge connecting to. If there is no such edge, then w(, ) =. 0 0 : Theory of lgorithms Spring 04 9 / 60 Single Source Shortest Paths Single Source Shortest Paths Single Source Shortest Paths Given start vertex s, find the shortest path from s to all other vertices. Try : Visit all vertices in some order, compute shortest paths for all vertices seen so far, then add the shortest path to next vertex x. Problem: Shortest path to a vertex already processed might go through x. Solution: Process vertices in order of distance from s. Given start vertex s, find the shortest path from s to all other vertices. Try : Visit all vertices in some order, compute shortest paths for all vertices seen so far, then add the shortest path to next vertex x. Problem: Shortest path to a vertex already processed might go through x. Solution: Process vertices in order of distance from s. : Theory of lgorithms Spring 04 0 / 60

6 0 0 ijkstra s lgorithm xample ijkstra s lgorithm xample ijkstra s lgorithm xample Initial 0 Process Process Process Process Process Initial 0 Process Process Process Process Process : Theory of lgorithms Spring 04 / 60 ijkstra s lgorithm: rray () void ijkstra(graph G, int s) { // Use array int [G.n()]; for (int i=0; i<g.n(); i++) // Initialize [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Process vertices int v = minvertex(g, ); if ([v] == ININITY) return; // Unreachable G.setMark(v, VISIT); for (dge w = each neighbor of v) if ([G.v(w)] > ([v] + G.weight(w))) [G.v(w)] = [v] + G.weight(w); ijkstra s lgorithm: rray () ijkstra s lgorithm: rray () void ijkstra(graph G, int s) { // Use array int [G.n()]; for (int i=0; i<g.n(); i++) // Initialize [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Process vertices int v = minvertex(g, ); if ([v] == ININITY) return; // Unreachable G.setMark(v, VISIT); for (dge w = each neighbor of v) if ([G.v(w)] > ([v] + G.weight(w))) [G.v(w)] = [v] + G.weight(w); : Theory of lgorithms Spring 04 / 60 ijkstra s lgorithm: rray () ijkstra s lgorithm: rray () ijkstra s lgorithm: rray () // Get mincost vertex int minvertex(graph G, int* ) { int v; // Initialize v to an unvisited vertex; for (int i=0; i<g.n(); i++) if (G.getMark(i) == UNVISIT) { v = i; break; for (i++; i<g.n(); i++) // ind smallest val if ((G.getMark(i)==UNVISIT) && ([i]<[v])) v = i; return v; // Get mincost vertex int minvertex(graph G, int* ) { int v; // Initialize v to an unvisited vertex; for (int i=0; i<g.n(); i++) if (G.getMark(i) == UNVISIT) { v = i; break; for (i++; i<g.n(); i++) // ind smallest val if ((G.getMark(i)==UNVISIT) && ([i]<[v])) v = i; return v; pproach : Scan the table on each pass for closest vertex. Total cost: Θ( V + ) = Θ( V ). pproach : Scan the table on each pass for closest vertex. Total cost: Θ( V + ) = Θ( V ). : Theory of lgorithms Spring 04 / 60 ijkstra s lgorithm: Priority Queue () class lem { public: int vertex, dist; ; int key(lem x) { return x.dist; void ijkstra(graph G, int s) { // priority queue int v; lem temp; int [G.n()]; lem [G.e()]; temp.dist = 0; temp.vertex = s; [0] = temp; heap H(,, G.e()); // reate the heap for (int i=0; i<g.n(); i++) [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Get distances do { temp = H.removemin(); v = temp.vertex; while (G.getMark(v) == VISIT); G.setMark(v, VISIT); if ([v] == ININITY) return; // Unreachable ijkstra s lgorithm: Priority Queue () ijkstra s lgorithm: Priority Queue () class lem { public: int vertex, dist; ; int key(lem x) { return x.dist; void ijkstra(graph G, int s) { // priority queue int v; lem temp; int [G.n()]; lem [G.e()]; temp.dist = 0; temp.vertex = s; [0] = temp; heap H(,, G.e()); // reate the heap for (int i=0; i<g.n(); i++) [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Get distances do { temp = H.removemin(); v = temp.vertex; while (G.getMark(v) == VISIT); G.setMark(v, VISIT); if ([v] == ININITY) return; // Unreachable : Theory of lgorithms Spring 04 4 / 60

7 ijkstra s lgorithm: Priority Queue () ijkstra s lgorithm: Priority Queue () ijkstra s lgorithm: Priority Queue () for (dge w = each neighbor of v) if ([G.v(w)] > ([v] + G.weight(w))) { [G.v(w)] = [v] + G.weight(w); temp.dist = [G.v(w)]; temp.vertex = G.v(w); H.insert(temp); // Insert new distance pproach : Store unprocessed vertices using a min-heap to implement a priority queue ordered by value. Must update priority queue for each edge. Total cost: Θ(( V + ) log V ). for (dge w = each neighbor of v) if ([G.v(w)] > ([v] + G.weight(w))) { [G.v(w)] = [v] + G.weight(w); temp.dist = [G.v(w)]; temp.vertex = G.v(w); H.insert(temp); // Insert new distance pproach : Store unprocessed vertices using a min-heap to implement a priority queue ordered by value. Must update priority queue for each edge. Total cost: Θ(( V + ) log V ). : Theory of lgorithms Spring 04 / 60 ll Pairs Shortest Paths or every vertex u, v V, calculate d(u, v). ould run ijkstra s lgorithm V times. etter is loyd s lgorithm. efine a k-path from u to v to be any path whose intermediate vertices all have indices less than k ll Pairs Shortest Paths Multiple runs of ijkstra s algorithm ost: V log V = V log V for dense graph. ll Pairs Shortest Paths or every vertex u, v V, calculate d(u, v). ould run ijkstra s lgorithm V times. etter is loyd s lgorithm. efine a k-path from u to v to be any path whose intermediate vertices all have indices less than k. The issue driving the concept of k paths is how to efficiently check all the paths without computing any path more than once. 0, is a 0-path.,0, is a -path. 0,, is a -path, but not a or path. verything is a 4 path. : Theory of lgorithms Spring 04 6 / 60 loyd s lgorithm loyd s lgorithm loyd s lgorithm void loyd(graph G) { // ll-pairs shortest paths int [G.n()][G.n()]; // Store distances for (int i=0; i<g.n(); i++) // Initialize for (int j=0; j<g.n(); j++) [i][j] = G.weight(i, j); for (int k=0; k<g.n(); k++) // ompute k paths for (int i=0; i<g.n(); i++) for (int j=0; j<g.n(); j++) if ([i][j] > ([i][k] + [k][j])) [i][j] = [i][k] + [k][j]; void loyd(graph G) { // ll-pairs shortest paths int [G.n()][G.n()]; // Store distances for (int i=0; i<g.n(); i++) // Initialize for (int j=0; j<g.n(); j++) [i][j] = G.weight(i, j); for (int k=0; k<g.n(); k++) // ompute k paths for (int i=0; i<g.n(); i++) for (int j=0; j<g.n(); j++) if ([i][j] > ([i][k] + [k][j])) [i][j] = [i][k] + [k][j]; : Theory of lgorithms Spring 04 7 / 60 Minimum ost Spanning Trees Minimum ost Spanning Trees Minimum ost Spanning Trees Minimum ost Spanning Tree (MST) Problem: Input: n undirected, connected graph G. Output: The subgraph of G that has minimum total cost as measured by summing the values for all of the edges in the subset, and keeps the vertices connected. Minimum ost Spanning Tree (MST) Problem: Input: n undirected, connected graph G. Output: The subgraph of G that has minimum total cost as measured by summing the values for all of the edges in the subset, and keeps the vertices connected : Theory of lgorithms Spring 04 8 / 60

8 Marked Vertices v i, i < j vu vp correct edge e ej Prim s edge Unmarked Vertices v, i >= j i vu vj Key Theorem for MST Let V, V be an arbitrary, non-trivial partition of V. Let (v, v ), v V, v V, be the cheapest edge between V and V. Then (v, v ) is in some MST of G. Proof: Let T be an arbitrary MST of G. If (v, v ) is in T, then we are done. Otherwise, adding (v, v ) to T creates a cycle. t least one edge (u, u ) of other than (v, v ) must be between V and V. c(u, u ) c(v, v ). Let T = T {(v, v ) {(u, u ). Then, T is a spanning tree of G and c(t ) c(t ). ut c(t ) is minimum cost. Therefore, c(t ) = c(t ) and T is a MST containing (v, v ). : Theory of lgorithms Spring 04 9 / 60 Key Theorem for MST Key Theorem for MST Let V, V be an arbitrary, non-trivial partition of V. Let (v, v), v V, v V, be the cheapest edge between V and V. Then (v, v) is in some MST of G. Proof: Let T be an arbitrary MST of G. If (v, v) is in T, then we are done. Otherwise, adding (v, v) to T creates a cycle. t least one edge (u, u) of other than (v, v) must be between V and V. c(u, u) c(v, v). Let T = T {(v, v) {(u, u). Then, T is a spanning tree of G and c(t ) c(t ). ut c(t ) is minimum cost. Therefore, c(t ) = c(t ) and T is a MST containing (v, v). There can only be multiple MSTs when there are edges with equal cost. Key Theorem igure Key Theorem igure Key Theorem igure Marked Vertices v i, i < j vu correct edge e Unmarked Vertices v, i >= j i vu vp ej Prim s edge vj : Theory of lgorithms Spring 04 0 / 60 Prim s MST lgorithm () void Prim(Graph G, int s) { // Prim s MST alg int [G.n()]; int V[G.n()]; // istances for (int i=0; i<g.n(); i++) // Initialize [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Process vertices int v = minvertex(g, ); G.setMark(v, VISIT); if (v!= s) dddgetomst(v[v], v); if ([v] == ININITY) return; //v unreachable for (dge w = each neighbor of v) if ([G.v(w)] > G.weight(w)) { [G.v(w)] = G.weight(w); // Update dist V[G.v(w)] = v; // who came from Prim s MST lgorithm () Prim s MST lgorithm () void Prim(Graph G, int s) { // Prim s MST alg int [G.n()]; int V[G.n()]; // istances for (int i=0; i<g.n(); i++) // Initialize [i] = ININITY; [s] = 0; for (i=0; i<g.n(); i++) { // Process vertices int v = minvertex(g, ); G.setMark(v, VISIT); if (v!= s) dddgetomst(v[v], v); if ([v] == ININITY) return; //v unreachable for (dge w = each neighbor of v) if ([G.v(w)] > G.weight(w)) { [G.v(w)] = G.weight(w); // Update dist V[G.v(w)] = v; // who came from : Theory of lgorithms Spring 04 / 60 Prim s MST lgorithm () Prim s MST lgorithm () Prim s MST lgorithm () int minvertex(graph G, int* ) { int v; // Initialize v to any unvisited vertex for (int i=0; i<g.n(); i++) if (G.getMark(i) == UNVISIT) { v = i; break; for (i=0; i<g.n(); i++) // ind smallest value if ((G.getMark(i)==UNVISIT) && ([i]<[v])) v = i; return v; This is an example of a greedy algorithm. int minvertex(graph G, int* ) { int v; // Initialize v to any unvisited vertex for (int i=0; i<g.n(); i++) if (G.getMark(i) == UNVISIT) { v = i; break; for (i=0; i<g.n(); i++) // ind smallest value if ((G.getMark(i)==UNVISIT) && ([i]<[v])) v = i; return v; This is an example of a greedy algorithm. : Theory of lgorithms Spring 04 / 60

9 lternative Prim s Implementation () lternative Prim s Implementation () lternative Prim s Implementation () Like ijkstra s algorithm, can implement with priority queue. void Prim(Graph G, int s) { int v; // The current vertex int [G.n()]; // istance array int V[G.n()]; // Who s closest lem temp; lem [G.e()]; // Heap array temp.distance = 0; temp.vertex = s; [0] = temp; // Initialize heap array heap H(,, G.e()); // reate the heap for (int i=0; i<g.n(); i++) [i] = ININITY; [s] = 0; Like ijkstra s algorithm, can implement with priority queue. void Prim(Graph G, int s) { int v; // The current vertex int [G.n()]; // istance array int V[G.n()]; // Who s closest lem temp; lem [G.e()]; // Heap array temp.distance = 0; temp.vertex = s; [0] = temp; // Initialize heap array heap H(,, G.e()); // reate the heap for (int i=0; i<g.n(); i++) [i] = ININITY; [s] = 0; : Theory of lgorithms Spring 04 / 60 lternative Prim s Implementation () for (i=0; i<g.n(); i++) { // Now build MST do { temp = H.removemin(); v = temp.vertex; while (G.getMark(v) == VISIT); G.setMark(v, VISIT); if (v!= s) dddgetomst(v[v], v); if ([v] == ININITY) return; // Unreachable for (dge w = each neighbor of v) if ([G.v(w)] > G.weight(w)) { // Update [G.v(w)] = G.weight(w); V[G.v(w)] = v; // Who came from temp.distance = [G.v(w)]; temp.vertex = G.v(w); H.insert(temp); // Insert dist in heap lternative Prim s Implementation () lternative Prim s Implementation () for (i=0; i<g.n(); i++) { // Now build MST do { temp = H.removemin(); v = temp.vertex; while (G.getMark(v) == VISIT); G.setMark(v, VISIT); if (v!= s) dddgetomst(v[v], v); if ([v] == ININITY) return; // Unreachable for (dge w = each neighbor of v) if ([G.v(w)] > G.weight(w)) { // Update [G.v(w)] = G.weight(w); V[G.v(w)] = v; // Who came from temp.distance = [G.v(w)]; temp.vertex = G.v(w); H.insert(temp); // Insert dist in heap : Theory of lgorithms Spring 04 4 / 60 Kruskal s MST lgorithm () Kruskal s MST lgorithm () Kruskal s MST lgorithm () Kruskel(Graph G) { // Kruskal s MST algorithm Gentree (G.n()); // quivalence class array lem [G.e()]; // rray of edges for min-heap int edgecnt = 0; for (int i=0; i<g.n(); i++) // Put edges into for (dge w = G.first(i); G.isdge(w); w = G.next(w)) { [edgecnt].weight = G.weight(w); [edgecnt++].edge = w; heap H(, edgecnt, edgecnt); // Heapify edges int nummst = G.n(); // Init w/ n equiv classes Kruskel(Graph G) { // Kruskal s MST algorithm Gentree (G.n()); // quivalence class array lem [G.e()]; // rray of edges for min-heap int edgecnt = 0; for (int i=0; i<g.n(); i++) // Put edges into for (dge w = G.first(i); G.isdge(w); w = G.next(w)) { [edgecnt].weight = G.weight(w); [edgecnt++].edge = w; heap H(, edgecnt, edgecnt); // Heapify edges int nummst = G.n(); // Init w/ n equiv classes : Theory of lgorithms Spring 04 / 60 Kruskal s MST lgorithm () Kruskal s MST lgorithm () Kruskal s MST lgorithm () for (i=0; nummst>; i++) { // ombine lem temp = H.removemin(); // Next cheap edge dge w = temp.edge; int v = G.v(w); int u = G.v(w); if (.differ(v, u)) { // If different.union(v, u); // ombine dddgetomst(g.v(w), G.v(w)); // dd nummst--; // Now one less MST for (i=0; nummst>; i++) { // ombine lem temp = H.removemin(); // Next cheap edge dge w = temp.edge; int v = G.v(w); int u = G.v(w); if (.differ(v, u)) { // If different.union(v, u); // ombine dddgetomst(g.v(w), G.v(w)); // dd nummst--; // Now one less MST How do we compute function MSTof(v)? Solution: UNION-IN algorithm (Section 4.). How do we compute function MSTof(v)? Solution: UNION-IN algorithm (Section 4.). : Theory of lgorithms Spring 04 6 / 60

10 Initial Step Process edge (, ) Step Process edge (, ) Step Process edge (, ) Kruskal s lgorithm xample Total cost: Θ( V + log ). Initial Step Process edge (, ) Kruskal s lgorithm xample Kruskal s lgorithm xample Total cost: Θ( V + log ). ost is dominated by the edge sort. lternative: Use a min heap, quit when only one set left. Kth-smallest implementation. Step Process edge (, ) Step Process edge (, ) : Theory of lgorithms Spring 04 7 / 60 Matching Suppose there are n workers that we want to work in teams of two. Only certain pairs of workers are willing to work together. Problem: orm as many compatible non-overlapping teams as possible. Model using G, an undirected graph. Join vertices if the workers will work together. matching is a set of edges in G with no vertex in more than one edge (the edges are independent). maximal matching has no free pairs of vertices that can extend the matching. maximum matching has the greatest possible number of edges. perfect matching includes every vertex. Matching Matching Suppose there are n workers that we want to work in teams of two. Only certain pairs of workers are willing to work together. Problem: orm as many compatible non-overlapping teams as possible. Model using G, an undirected graph. Join vertices if the workers will work together. matching is a set of edges in G with no vertex in more than one edge (the edges are independent). maximal matching has no free pairs of vertices that can extend the matching. maximum matching has the greatest possible number of edges. perfect matching includes every vertex. n example: (-) is a matching. (-) (, 4) is both maximal and maximum. Take away the edge (-4). Then (, ) would be maximal but not a maximum matching. 4 : Theory of lgorithms Spring 04 8 / 60 Very ense Graphs () Theorem: Let G = (V, ) be an undirected graph with V = n and every vertex having degree n. Then G contains a perfect matching. Proof: Suppose that G does not contain a perfect matching. Let M be a max matching. M < n. There must be two unmatched vertices v, v that are not adjacent. very vertex adjacent to v or to v is matched. Let M M be the set of edges involved in matching the neighbors of v and v. There are n edges from v and v to vertices covered by M, but M < n. Very ense Graphs () Very ense Graphs () Theorem: Let G = (V, ) be an undirected graph with V = n and every vertex having degree n. Then G contains a perfect matching. Proof: Suppose that G does not contain a perfect matching. Let M be a max matching. M < n. There must be two unmatched vertices v, v that are not adjacent. very vertex adjacent to v or to v is matched. Let M M be the set of edges involved in matching the neighbors of v and v. There are n edges from v and v to vertices covered by M, but M < n. There must be two unmatched vertices not adjacent: Otherwise it would either be perfect (if there are no free vertices) or we could just match v and v (because they are adjacent). very adjacent vertex is matched, otherwise the matching would not be maximal. See Manber igure.76. : Theory of lgorithms Spring 04 9 / 60 Very ense Graphs () Very ense Graphs () Very ense Graphs () Proof: (continued) Thus, some edge of M is adjacent to edges from v and v. Let (u, u) be such an edge. Replacing (u, u) with (v, u) and (v, u) results in a larger matching. Theorem proven by contradiction. Proof: (continued) Thus, some edge of M is adjacent to edges from v and v. Let (u, u ) be such an edge. Replacing (u, u ) with (v, u ) and (v, u ) results in a larger matching. Theorem proven by contradiction. Pigeonhole Principle : Theory of lgorithms Spring / 60

11 v v v u u u v u Generalizing the Insight Generalizing the Insight Generalizing the Insight v, u, u, v is a path from an unmatched vertex to an unmatched vertex such that alternate edges are unmatched and matched. In one step, switch unmatched and matched edges. Let G = (V, ) be an undirected graph and M a matching. n alternating path P goes from v to u, consists of v v v v alternately matched and unmatched edges, and both v and u are not in the match. u u u u v, u, u, v is a path from an unmatched vertex to an unmatched vertex such that alternate edges are unmatched and matched. In one step, switch unmatched and matched edges. Let G = (V, ) be an undirected graph and M a matching. n alternating path P goes from v to u, consists of alternately matched and unmatched edges, and both v and u are not in the match. : Theory of lgorithms Spring 04 4 / 60 Matching xample Matching xample Matching xample ,,, is NOT an alternating path (it does not start with an unmatch vertex). 7, 6,, 0, 9, 8 is an alternating path with respect to the given matching Observation: If a matching has an alternating path, then the size of the matching can be increased by one by switching matched and unmatched edges along the alternating path. 0 9 : Theory of lgorithms Spring 04 4 / 60 The lternating Path Theorem () Theorem: matching is maximum iff it has no alternating paths. Proof: learly, if a matching has alternating paths, then it is not maximum. Suppose M is a non-maximum matching. Let M be any maximum matching. Then, M > M. Let M M be the symmetric difference of M and M. M M = M M (M M ). G = (V, M M ) is a subgraph of G having maximum degree. The lternating Path Theorem () The lternating Path Theorem () Theorem: matching is maximum iff it has no alternating paths. Proof: learly, if a matching has alternating paths, then it is not maximum. Suppose M is a non-maximum matching. Let M be any maximum matching. Then, M > M. Let M M be the symmetric difference of M and M. M M = M M (M M ). G = (V, M M ) is a subgraph of G having maximum degree. The first point is the obvious part of the iff. If there is an alternating path, simply switch the match and umatched edges to augment the match. Symmetric difference: Those in either, but not both. The max degree is because a vertex matches one different vertex in M and M. : Theory of lgorithms Spring 04 4 / 60 The lternating Path Theorem () The lternating Path Theorem () The lternating Path Theorem () Proof: (continued) Therefore, the connected components of G are either even-length cycles or a path with alternating edges. Since M > M, there must be a component of G that is an alternating path having more M edges than M edges. Proof: (continued) Therefore, the connected components of G are either even-length cycles or a path with alternating edges. Since M > M, there must be a component of G that is an alternating path having more M edges than M edges. : Theory of lgorithms Spring / 60

12 ipartite Matching bipartite graph G = (U, V, ) consists of two disjoint sets of vertices U and V together with edges such that every edge has an endpoint in U and an endpoint in V. ipartite matching naturally models a number of assignment problems, such as assignment of workers to jobs. lternating paths will work to find a maximum bipartite matching. n alternating path always has one end in U and the other in V. If we direct unmatched edges from U to V and matched edges from V to U, then a directed path from an unmatched vertex in U to an unmatched vertex in V is an alternating path. : Theory of lgorithms Spring 04 4 / 60 ipartite Matching ipartite Matching bipartite graph G = (U, V, ) consists of two disjoint sets of vertices U and V together with edges such that every edge has an endpoint in U and an endpoint in V. ipartite matching naturally models a number of assignment problems, such as assignment of workers to jobs. lternating paths will work to find a maximum bipartite matching. n alternating path always has one end in U and the other in V. If we direct unmatched edges from U to V and matched edges from V to U, then a directed path from an unmatched vertex in U to an unmatched vertex in V is an alternating path. ipartite Matching xample ipartite Matching xample ipartite Matching xample, 8,, 0 is an alternating path. 4, 6,, 7, 4, 9 and, 8,, 0 are disjoint alternating paths that we can augment independently. Naive algorithm: ind a maximal matching (greedy algorithm). or each vertex: o a S or other search until an alternating path is found. Use the alternating path to improve the match , 8,, 0 is an alternating path. 0 V ( V + ), 6,, 7, 4, 9 and, 8,, 0 are disjoint alternating paths that we can augment independently. : Theory of lgorithms Spring / 60 lgorithm for Maximum ipartite Matching onstruct S subgraph from the set of unmatched vertices in U until a level with unmatched vertices in V is found. lgorithm for Maximum ipartite Matching lgorithm for Maximum ipartite Matching onstruct S subgraph from the set of unmatched vertices in U until a level with unmatched vertices in V is found. Greedily select a maximal set of disjoint alternating paths. ugment along each path independently. Repeat until no alternating paths remain. Time complexity O(( V + ) V ). Order doesn t matter. ind a path, remove its vertices, then repeat.ugment along the paths independently since they are disjoint. Greedily select a maximal set of disjoint alternating paths. ugment along each path independently. Repeat until no alternating paths remain. Time complexity O(( V + ) V ). : Theory of lgorithms Spring / 60