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CS 253: Algorithms

CS 253: Algorithms. Chapter 23 Minimum Spanning Tree. Credit : Dr. George Bebis. 8. 7. b. c. d. 9. 4. 2. a. e. i. 11. 14. 4. 6. 7. 8. 10. h. g. f. 2. 1. Minimum Spanning Trees. Spanning Tree

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CS 253: Algorithms

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  1. CS 253: Algorithms Chapter 23 Minimum Spanning Tree Credit: Dr. George Bebis

  2. 8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 Minimum Spanning Trees • Spanning Tree • A tree (i.e., connected, acyclic graph) which contains all the vertices of the graph • Minimum Spanning Tree • Spanning tree with the minimum sum of weights • Spanning forest • If a graph is not connected, then there is a spanning tree for each connected component of the graph

  3. 8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 Sample applications of MST • Find the least expensive way to connect a set of cities, terminals, computers, etc. • A town has a set of houses. A road connects 2 and only 2 houses. Repair Cost for road (u,v) is w(u, v) Problem: Repair enough roads such that: • Everyone stays connected • Total repair cost is minimum

  4. Properties of Minimum Spanning Trees • Minimum spanning tree is not unique • MST has no cycles (by definition) : • # of edges in a MST: |V| - 1

  5. Prim’s Algorithm • Starts from an arbitrary “root”: VA = {a} • At each step: • Find a light edge crossing (VA, V - VA) • Add this edge to set A (The edges in set A always form a single tree) • Repeat until the tree spans all vertices 8 7 b c d 4 9 2 a e i 11 14 4 6 7 8 10 h g f 2 1

  6. 8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 How to Find Light Edges Quickly? • Use a priority queue Q contains vertices not yet included in the tree, i.e. (V - VA) VA= {a}, Q = {b, c, d, e, f, g, h, i} • We associate a key with each vertex v in Q: key[v] = minimum weight of any edge (u, v)connecting v to VA • After adding a new node to VA we update the weights of all the nodes adjacent to it. e.g., after adding a to the tree, Key[b]=4 and Key[h]=8

  7.        8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 Example 0         Q = {a, b, c, d, e, f, g, h, i} VA =  Extract-MIN(Q)  a 4 key [b] = 4  [b] = a key [h] = 8  [h] = a 4      8  Q = {b, c, d, e, f, g, h, i} VA = {a} Extract-MIN(Q)  b    8 7 b c d 4 9  2  a e i 11 14 4 6 7 8 10 h g f 2 1    8

  8. 4    8 8 8 7 7 b b c c d d 8  4 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4   6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1   8 Example Q = {c, d, e, f, g, h, i} VA = {a, b} key [c] = 8 [c] = b key [h] = 8  [h] = a - unchanged 8     8  Extract-MIN(Q)  c 8   Q = {d, e, f, g, h, i} VA = {a, b, c} key [d] = 7 [d] = c key [f] = 4  [f] = c key [i] = 2 [i] = c 7  4  8 2 Extract-MIN(Q)  i 7 2 4

  9. 8 8 7 7 8 b b c c d d 4 7 8 4 7 9 9 4 4 2 2 a a e e i i 11 11 14 14 2 4 4   2 6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1 4  8 4 6 7 Example Q = {d, e, f, g, h} VA = {a, b, c, i} key [h] = 7 [h] = i key [g] = 6  [g] = i 7  4 6 7 Extract-MIN(Q)  f 6 7 Q = {d, e, g, h} VA = {a, b, c, i, f} key [g] = 2 [g] = f key [d] = 7  [d] = c unchanged key [e] = 10  [e] = f 7 10 2 7 Extract-MIN(Q)  g 10 2

  10. 8 8 7 7 b b c c d d 8 8 4 4 7 7 9 9 4 4 2 2 a a e e i i 11 11 14 14 4 4 10 10 2 2 6 6 7 7 8 8 10 10 h h g g f f 2 2 1 1 4 4 2 2 7 1 Example Q = {d, e, h} VA = {a, b, c, i, f, g} key [h] = 1 [h] = g 7 10 1 Extract-MIN(Q)  h 1 Q = {d, e} VA = {a, b, c, i, f, g, h} 7 10 Extract-MIN(Q)  d

  11. 8 7 b c d 8 4 7 9 4 2 a e i 11 14 4 10 2 6 7 8 10 h g f 2 1 4 2 1 Example Q = {e} VA = {a, b, c, i, f, g, h, d} key [e] = 9  [e] = d 9 Extract-MIN(Q)  e Q =  VA = {a, b, c, i, f, g, h, d, e} 9

  12. PRIM(V, E, w, r) % r : starting vertex Total time: O(VlgV+ ElgV) = O(ElgV) • Q ←  • for each u V • do key[u] ← ∞ • π[u] ← NIL • INSERT(Q, u) • DECREASE-KEY(Q, r, 0) % key[r] ← 0 • while Q   • do u ← EXTRACT-MIN(Q) • for each vAdj[u] • do if v  Q and w(u, v) < key[v] • then π[v] ← u • DECREASE-KEY(Q, v, w(u, v)) O(V)if Q is implemented as a min-heap O(lgV) Min-heap operations: O(VlgV) Executed |V| times Takes O(lgV) Executed O(E) times total Constant Takes O(lgV) O(ElgV)

  13. Prim’s Algorithm • Total time: O(ElgV ) • Prim’s algorithm is a “greedy” algorithm • Greedy algorithms find solutions based on a sequence of choices which are “locally” optimal at each step. • Nevertheless, Prim’s greedy strategy produces a globally optimum solution!

  14. We would add edge (c, f) 8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 Kruskal’s Algorithm • Start with each vertex being its own component • Repeatedly merge two components into one by choosing the lightestedge that connects them • Which components to consider at each iteration? • Scan the set of edges in monotonically increasing order by weight. Choose the smallest edge.

  15. 8 7 b c d 9 4 2 a e i 11 14 4 6 7 8 10 h g f 2 1 Example {g, h}, {a}, {b}, {c},{d},{e},{f},{i} {g, h}, {c, i}, {a}, {b}, {d}, {e}, {f} {g, h, f}, {c, i}, {a}, {b}, {d}, {e} {g, h, f}, {c, i}, {a, b}, {d}, {e} {g, h, f, c, i}, {a, b}, {d}, {e} {g, h, f, c, i}, {a, b}, {d}, {e} {g, h, f, c, i, d}, {a, b}, {e} {g, h, f, c, i, d}, {a, b}, {e} {g, h, f, c, i, d, a, b}, {e} {g, h, f, c, i, d, a, b}, {e} {g, h, f, c, i, d, a, b, e} {g, h, f, c, i, d, a, b, e} {g, h, f, c, i, d, a, b, e} {g, h, f, c, i, d, a, b, e} • Add (h, g) • Add (c, i) • Add (g, f) • Add (a, b) • Add (c, f) • Ignore (i, g) • Add (c, d) • Ignore (i, h) • Add (a, h) • Ignore (b, c) • Add (d, e) • Ignore (e, f) • Ignore (b, h) • Ignore (d, f) 1: (h, g) 2: (c, i), (g, f) 4: (a, b), (c, f) 6: (i, g) 7: (c, d), (i, h) 8: (a, h), (b, c) 9: (d, e) 10: (e, f) 11: (b, h) 14: (d, f) {a}, {b}, {c}, {d}, {e}, {f}, {g}, {h},{i}

  16. Operations on Disjoint Data Sets • Kruskal’s Alg. uses Disjoint Data Sets (UNION-FIND : Chapter 21) to determine whether an edge connects vertices in different components • MAKE-SET(u) – creates a new set whose only member is u • FIND-SET(u) – returns a representative element from the set that contains u. It returns the same value for any element in the set • UNION(u, v) – unites the sets that contain u and v, say Su and Sv • E.g.: Su = {r, s, t, u}, Sv= {v, x, y} UNION (u, v) = {r, s, t, u, v, x, y} • We had seen earlier that FIND-SET can be done in O(lgn) or O(1)time and UNION operation can be done in O(1) (see Chapter 21)

  17. KRUSKAL(V, E, w) O(V) • A ←  • for each vertex v V • do MAKE-SET(v) • sort E into non-decreasing order by w • for each (u, v) taken from the sorted list • do if FIND-SET(u)  FIND-SET(v) • then A ← A  {(u, v)} • UNION(u, v) • return A Running time: O(V+ElgE+ElgV)  O(ElgE) • Implemented by using the disjoint-set data structure (UNION-FIND) • Kruskal’salgorithm is “greedy” • It produces a globally optimum solution O(ElgE) O(E) O(lgV)

  18. Problem 1 Compare Prim’s algorithm with Kruskal’salgorithm assuming: (a) Sparse graphs:  E=O(V) Kruskal: UNION-FIND: O(ElgE) = O(VlgV) Prim: Binary heap: O(ElgV) = O(VlgV) (b) Dense graphs:  E = O(V2) Kruskal: O(ElgE) = O(V2lgV2) = O(2V2lgV) = O(V2lgV) Prim: Binary heap: O(ElgV) = O(V2lgV)

  19. Problem 2 • Analyze the running time of Kruskal’s algorithm when weights are in the range [1 … V] • ANSWER: • Sorting can be done in O(E) time (e.g., using counting sort) • However, overall running time will not change, i.e., O(ElgV)

  20. Problem 3 • Suppose that some of the weights in a connected graph G are negative. Will Prim’s algorithm still work? What about Kruskal’s algorithm? Justify your answers. ANSWER: Yes, both algorithms will work with negative weights. There is no assumption in the algorithm about the weights being positive.

  21. Problem 4 Analyze Prim’s algorithm assuming: (a) an adjacency-list representation of G O(ElgV) (b) an adjacency-matrix representation of G O(ElgV+V2) (see next slide)

  22. PRIM(V, E, w, r) Total time: O(VlgV+ ElgV) = O(ElgV) • Q ←  • for each u V • do key[u] ← ∞ • π[u] ← NIL • INSERT(Q, u) • DECREASE-KEY(Q, r, 0) % key[r] ← 0 • while Q   • do u ← EXTRACT-MIN(Q) • for each vAdj[u] • do if v  Q and w(u, v) < key[v] • then π[v] ← u • DECREASE-KEY(Q, v, w(u, v)) O(V)if Q is implemented as a min-heap O(lgV) Min-heap operations: O(VlgV) Executed |V| times Takes O(lgV) Executed O(E) times total Constant O(ElgV) Takes O(lgV)

  23. PRIM(V, E, w, r) Total time: O(ElgV + V2) • Q ←  • for each u V • do key[u] ← ∞ • π[u] ← NIL • INSERT(Q, u) • DECREASE-KEY(Q, r, 0) % key[r] ← 0 • while Q   • do u ← EXTRACT-MIN(Q) • for (j=0; j<|V|; j++) • do if (A[u][j]=1) and (vQ) and (w(u, v)<key[v]) • then π[v] ← u • DECREASE-KEY(Q, v, w(u, v)) O(V)if Q is implemented as a min-heap O(lgV) Min-heap operations: O(VlgV) Executed |V| times Takes O(lgV) Executed O(V2)times total O(ElgV) Takes O(lgV)

  24. Problem 5 • Find an algorithm for the “maximum” spanning tree. That is, given an undirected weighted graph G, find a spanning tree of G of maximum cost. Prove the correctness of your algorithm. • Consider choosing the “heaviest” edge (i.e., the edge associated with the largest weight) in a cut. The generic proof can be modified easily to show that this approach will work. • Alternatively, multiply the weights by -1 and apply either Prim’s or Kruskal’s algorithms without any modification at all!

  25. Problem 6 Let T be a MST of a graph G, and let L be the sorted list of the edge weights of T. Show that for any other MST T’ of G, the list L is also the sorted list of the edge weights of T’ Proof: Kruskal’s algorithm will find T in the order specified by L. Similarly, if T’ is also an MST, Kruskal’s algorithm should be able to find it in the sorted order, L’. If L’  L, then there is a contradiction! Because at the point where L and L’ differ, Kruskal’s should have picked the smaller of the two and therefore L’ is impossible to obtain. T, L = {1,2} T’, L’ = {1,2}

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