Greedy Algorithms Pasi Frnti 15 11 2017 Greedy

Greedy Algorithms Pasi Fränti 15. 11. 2017

Greedy algorithm 1. Coin problem 2. Minimum spanning tree 3. Generalized knapsack problem 4. Traveling salesman problem

Coin problem Task: Given a coin set, pay the required amount (36 snt) using least number of coins.

Coin problem Another coin set 25 10 1 Amount to be paid: 30 Greedy: Optimal:

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Minimum spanning tree When greedy works Input: Output: Graph Sub-graph Tree Spanning tree Minimum No cycles All nodes reachable Sum of weights minimum

Minimum spanning tree Prim’s algorithm Prim(V, E): RETURN T Select (u, v) E with min weight S S {u, v}; P P {(u, v)}; E E{(u, v)}; REPEAT Select (u, v) with min weight (u, v) E, u S, v S S S {v}; P P {(u, v)}; E E{(u, v)}; UNTIL S=V Return P;

Example of Prim 117 216 110 246 199 121 79 182 315 142 170 242 191 148 231 136 78 120 126 149 89 116 234 170 51 112 79 131 109 86 73 163 143 72 63 90 53 191 178 59 58 116 105 135 27

Proof of optimality General properties of spanning trees • Spanning tree includes N-1 links • There are no cycles • Minimum spanning tree is the one with the smallest weights Cycle C A B No cycle Remove Link BC C A B

Proof of optimality Case: minimum link C 2 A 1 B 2 C 2 Add AB • Link AB is minimum • Suppose it is not in MST • Path A→B must exist A 1 B 2 • Adding AB we can remove another link (e. g. AC) • Path A→C exists • All nodes reached from C can now be reached from B

Proof of optimality Induction step MST solved for Subset S 3 E C 4 A B D Replace CD by CE • Suppose CE is minimum connecting S outside • Path D→E must exist and is outside S

Proof of optimality Induction step MST solved for Subset S 3 E C 4 A B D • Path D→E still exist as before • All nodes reachable via D can now be reached via C→E→D

Sou rce o f da ta ju st fo r fun

Minimum spanning tree Kruskal’s algorithm Kruskal(V, E): RETURN T FOR ALL v V Create. Set(v); P ; SORT(E); // according to decreasing weights FOR each e E DO IF Find. Set(u) Find. Set(v) THEN UNION(u, v); P P {(u, v)}; Return P;

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Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Manchester 40 30 Liverpool Sheffield 110 70 40 Shrewsbury 50 50 80 Nottingham B/ham 110 Aberystwyth 70 100 90 120 50 Cardiff Oxford Bristol 80 Southampton 70

Further questions • • What is the time complexity of Prim? What should we use: array, list or heap? How about union-find structure? Nearest neighbor link solution better?

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Knapsack problem Problem definition Input: Weight of N items {w 1, w 2, . . . , wn} Cost of N items {c 1, c 2, . . . , cn} Knapsack limit S Output: Selection for knapsack: {x 1, x 2, …xn} where xi {0, 1}. Sample input: wi={1, 1, 2, 4, 12} ci ={1, 2, 2, 10, 4} S=15 Binary choice

Generalized Knapsack problem Input: Weight of N items {w 1, w 2, . . . , wn} Cost of N items {c 1, c 2, . . . , cn} Knapsack limit S Output: Selection for knapsack: {x 1, x 2, …xn} where xi [0, 1]. Any real value between 0 and 1 ! Sample input: wi={1, 1, 2, 4, 12} ci ={1, 2, 2, 10, 4} S=15 ci/wi = {1, 2, 1, 2. 5, 0. 33} Sorted = {2. 5, 2, 1, 1, 0. 33} Select = 1. 0, 0. 58 Costs Weights 10 4 2 1 2 2 1 1 2. 3 7. 0

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Traveling salesman problem Greedy. TSP(V, E, home): RETURN T For i 1 TO N DO Visited[i] FALSE; X[1] home; Visited[home] TRUE; Sub -op tim (Exe al! rcise task ) FOR i 1 TO N-1 DO Select (X[i], v) E weight is minimal X[2] v; Visited[v] TRUE;

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Greedy algorithms Not 1. Coin problem Solv 2. Minimum spanning tree ed Solv 3. Generalized knapsack problem ed Not 4. Traveling salesman problem