Undirected connected weighted graph ( G , W ) Input: An MST of G - PowerPoint PPT Presentation

Minimum Spanning Trees ( G , W ) undirected connected weighted graph Definition 11.1 A minimum spanning tree (MST) of G is a connected spanning subgraph T of G of minimum weight. The minimum spanning tree problem : Undirected connected weighted graph ( G , W ) Input: An MST of G Output: A&DS Lecture 11 1 Mary Cryan

Kruskal’s Algorithm A different approach to computing MSTs. A forest is a graph whose connected components are trees. Idea Starting from the spanning forest without any edges, repeatedly add edges of minimum weight until the forest becomes a tree. Algorithm K RUSKAL ( G , W ) 1. F ← ∅ 2. for all e ∈ E in the order of increasing weight do if the endpoints of e belong to different connected 3. components of ( V, F ) then F ← F ∪ { e } 4. 5. return tree with edge set F A&DS Lecture 11 2 Mary Cryan

Correctness of Kruskal’s algorithm 1. Throughout the execution of K RUSKAL , ( V, F ) remains a spanning forest. Proof: ( V, F ) is a spanning subgraph because the vertex set is V . It always remains a forest because edges with endpoints in different connected components never induce a cycle. 2. Eventually, ( V, F ) will be connected and thus a spanning tree. Proof: Suppose that after the complete execution of the loop, ( V, F ) has a connected component ( V 1 , F 1 ) with V 1 � = V . Since G is connected, there is an edge e ∈ E with exactly one endpoint in V 1 . This edge would have been added to F when being processed in the loop, so this can never happen. 3. Throughout the execution of K RUSKAL , ( V, F ) is contained in some MST of G . Proof: Similar to the proof of the corresponding statement for Prim’s algorithm. A&DS Lecture 11 3 Mary Cryan

Data Structures for Disjoint Sets • A disjoint set data structure maintains a collection S = { S 1 , . . . , S k } of disjoint sets . • The sets are dynamic , i.e., they may change over time. • Each set S i is identified by some representative , which is some member of that set. Operations: • M AKE -S ET ( x ) : Creates new set whose only member is x . The representative is x . • U NION ( x, y ) : Unites set S x containing x and set S y containing y into a new set S and removes S x and S y from the collection. • F IND -S ET ( x ) : Returns representative of the set holding x . A&DS Lecture 11 4 Mary Cryan

Implementation of Kruskal’s Algorithm Algorithm K RUSKAL ( G , W ) 1. F ← 0 2. for all vertices v of G do M AKE -S ET ( v ) 3. 4. sort edges of G into non-decreasing order by weight 5. for all edges ( u, v ) of G in non-decreasing order by weight do if F IND -S ET ( u ) � = F IND -S ET ( v ) then 6. F ← F ∪ { ( u, v ) } 7. U NION ( u, v ) 8. 9. return F A&DS Lecture 11 5 Mary Cryan

Analysis of K RUSKAL Let n be the number of vertices and m the number of edges of the input graph • Line 1, line 9: Θ ( 1 ) • Loop in Lines 2–3: Θ ( n · T M AKE -S ET ( n )) • Line 4 : Θ ( m lg m ) � � • Loop in Lines 5–8: Θ m · T F IND -S ET ( n ) + n · T U NION ( n ) . Overall: � �� � � � Θ n T M AKE -S ET ( n ) + T U NION ( n ) + m lg m + T F IND -S ET ( n ) With efficient implementation of disjoint set this amounts to T ( n, m ) = Θ ( m lg m ) . A&DS Lecture 11 6 Mary Cryan

Amortized Analysis We want to analyse the following: Given a sequence of m “Disjoint Set” operations (M AKE -S ET , U NION , F IND -S ET ), bound the total running time to perform a sequence of m operations . • Usually parametrize in terms of n (no. of M AKE -S ET operations), not just m . • (Of course) the bound we get will depend on the particular data structure/implementation we design for this problem. A&DS Lecture 11 7 Mary Cryan

Linked List Implementation of Disjoint Sets Each element represented by a pointer to a cell: x Use a linked list for each set. Representative of the set is at the head of the list. Each cell has a pointer direct to the representative (head of the list). A&DS Lecture 11 8 Mary Cryan

Example Linked list representation of { a, f } , { b } , { g, c, e } , { d } : a f b g c e d The representatives are a , b , g and d . A&DS Lecture 11 9 Mary Cryan

Analysis of Linked List Implementation M AKE -S ET : constant time. F IND -S ET : constant time. U NION : Naive implementation of U NION ( x, y ) appends x ’s list onto end of y ’s list. Assumption: Representative y of each set has attribute last [ y ] : a pointer to last cell of y ’s list. (Not shown in diagrams.) Snag: have to update “representative pointer” in each cell of x ’s list to point to the representative (head) of y ’s list. Cost is: Θ ( length of x ’s list ) . A&DS Lecture 11 10 Mary Cryan

Example (cont’d) a f b g c e d U NION ( g, b ) A&DS Lecture 11 11 Mary Cryan

Conventions for Further Analysis Express running time in terms of: n : the number of M AKE -S ET operations, m : the number of M AKE -S ET , U NION and F IND -S ET operations. Note 1. After n − 1 U NION operations only one set remains. 2. m ≥ n . A&DS Lecture 11 12 Mary Cryan

A nasty example Take: n = ⌈ m/2 ⌉ + 1 , q = m − n = ⌊ m/2 ⌋ − 1 . Elements: x 1 ,x 2 ,...,x n . Operation Number of objects updated M AKE -S ET ( x 1 ) 1 M AKE -S ET ( x 2 ) 1 . . . . . . M AKE -S ET ( x n ) 1 U NION ( x 1 , x 2 ) 1 U NION ( x 2 , x 3 ) 2 U NION ( x 3 , x 4 ) 3 . . . . . . U NION ( x q − 1 , x q ) q − 1 Θ ( m 2 ) Total A&DS Lecture 11 13 Mary Cryan

The Weighted-Union Heuristics Idea Record length of each list. To execute U NION ( x, y ) append shorter list to longer one (breaking ties arbitrarily). Theorem 11.2 Using the linked-list representation of disjoint sets and the weighted-union heuristic, a sequence of m M AKE -S ET , U NION & F IND -S ET operations, n of which are M AKE -S ET operations, takes O ( m + n lg n ) time. Crucial Proof Idea: Each element can appear at most lg n times in the shorter list of a U NION . (because if x is in the shortest list, the UNION operation doubles the size of x ’s list). A&DS Lecture 11 14 Mary Cryan

The Forest Implementation of Disjoint-Sets Each set represented by a rooted tree: g b i e a f h c d A&DS Lecture 11 15 Mary Cryan

Basic Operations M AKE -S ET : Constant time. F IND -S ET : Follow pointers to root. (Path followed is called the find path .) Cost proportional to height of tree. U NION : Naive strategy: root of tree of x made to point to that of y . Cost proportional to height of x ’s tree plus the height of y ’s tree. Not faster than linked list implementation. A&DS Lecture 11 16 Mary Cryan

Improving the Running Time General Strategy Keep trees low. Two Heuristics 1. Union-by-Rank: Attach lower tree to the root of higher tree in U NION 2. Path Compression: Update tree during F IND -S ET operations. A&DS Lecture 11 17 Mary Cryan

The Union-by-Rank Heuristic At each node x maintain a variable rank [ x ] , such that if x is the root of its tree, rank [ x ] is the height of this tree. In executing U NION ( x, y ) make the tree of smaller rank point to the one with larger rank. (Break ties arbitrarily.) A&DS Lecture 11 18 Mary Cryan

g f b d rank [ f ] = 0 rank [ d ] = 0 a e c U NION ( f, d ) rank [ g ] = 1 rank [ d ] = 1 g b d e c a f U NION ( f, g ) rank [ g ] = 2 g b e c a d f A&DS Lecture 11 19 Mary Cryan

The Height of the Trees Lemma 11.3 The height of a tree (in our datastructure) is at most lg ( size of the tree ) . Proof By induction on the number of U NION operations it is proved that a tree of height h has size at least 2 h . • As long as there are no U NION s, all trees have height 0 and contain 1 = 2 0 node. • Suppose U NION ( x,y ) is executed. Let r x and r y be the roots of the trees of x and y , resp. Case 1: rank ( r x ) < rank ( r y ) . Then r x becomes child of r y , and the height remains unchanged, but the number of nodes increases. Case 2: rank ( r x ) > rank ( r y ) . Analogously. Case 3: rank ( r x ) = rank ( r y ) . Then height increase by one and the size of the new tree is size ( r x ) + size ( r y ) ≥ 2 rank ( r x ) + 2 rank ( r y ) = 2 rank ( r x )+ 1 , which is the height of the new tree.

Running Time with Union-by-Rank Only M AKE -S ET : constant time. F IND -S ET : Bounded by rank: O ( log n ) . U NION : Bounded by rank: O ( log n ) . Bottom line ( m operations, of which n are M AKE -S ET ): O ( m log n ) . NOT any better than Linked-Lists with Weighted union heuristic. A&DS Lecture 11 21 Mary Cryan

The Path-Compression Heuristics Idea When performing F IND -S ET ( x ) make each vertex on find path point to root. Implementation Algorithm F IND -S ET ( x ) 1. if x � = π ( x ) then π ( x ) ← F IND -S ET ( π ( x )) 2. 3. return π ( x ) A&DS Lecture 11 22 Mary Cryan

Example f e d f c c a d b e b a F IND -S ET ( a ) A&DS Lecture 11 23 Mary Cryan

The Ackermann Function Ackerman’s Function Function A : N × N → N defined by 2 j for j ≥ 1 A ( 1, j ) = for i ≥ 2 A ( i, 1 ) = A ( i − 1, 2 ) A ( i, j ) = A ( i − 1, A ( i, j − 1 )) for i, j ≥ 2 Other variants exist—last line of definition is crucial common point. Inverse Not true mathematical inverse—grows as slowly as A ( i, j ) grows fast : α ( m, n ) = min { i ≥ 1 | A ( i, ⌊ m/n ⌋ ) > lg n } . A&DS Lecture 11 24 Mary Cryan