The Algorithms of Prim and Dijkstra. Class 19

Transcription

1 The Algorithms of Prim and Dijkstra Class 19

2 MST Introduction you live in a county that has terrible roads a new county commissioner was elected based on her campaign promise to repave the roads she promised to enable a citizen to travel from any town to any town in the entire county using only repaved roads she has just hired you to figure out how to keep her campaign promise for the least amount of money

3 The County the graph represents a map of the county A through F are towns, the edges are roads a weight represents the cost in millions to repave that road A 4 B C 1 D E F

4 Spanning Tree a spanning tree T of a connected undirected graph G is the set of all vertices V of G and a subset of the edges E of G such that T is a tree that contains all vertices of G if G is a weighted graph, then a minimal spanning tree is a spanning tree of minimal total edge weight the word minimal means finding an MST is an optimization problem

5 MST Algorithm there are two well-known greedy strategies for accomplishing this step Kruskal s Algorithm uses a data structure called disjoint sets Prim s Algorithm typically uses a priority queue we will only consider Prim s, because we know about PQs both algorithms are greedy

6 Prim s Algorithm Prim s algorithm assumes an undirected connected weighted graph it begins with a specified start vertex and no edges the greedy rule: add an edge of minimum weight that does not generate a cycle

7 Prim s Algorithm Prim s algorithm assumes an undirected connected weighted graph it begins with a specified start vertex and no edges the greedy rule: add an edge of minimum weight that does not generate a cycle note also that the PQ used by Prim s is built on a min-heap: the smallest value is at the root of the heap

8 An Example A 4 B C 1 D E 2 F

9 An Example A 4 B C 1 D E 2 F A B C D E F B:4 C:2 E:3 A:4 D:5 A:2 D:1 E:6 F:3 B:5 C:1 F:6 A:3 C:6 F:2 C:3 D:6 E:2

10 Prim s Algorithm void prim(graph, start, parent) { parent.at(start) = start; populate pq (start = 0, all other vertices ); for (i in 0.. n - 1) { vertex = pq.top(); pq.pop(); foreach w adjacent to vertex { if (pq.has_key(w) && graph_weight(vertex:w) < pq.get_weight(w)) { parent.at(w) = vertex; pq.change_weight(w, graph_weight(vertex:w)); } } } }

11 Change Key Prim s algorithm requires two new functions for pq: change weight and has key change weight changes the weight value for an entry in the pq and then calls bubble up or percolate down as necessary to readjust has key returns true or false for whether this key is in the PQ can also be implemented as find key

12 Change Key Prim s algorithm requires two new functions for pq: change weight and has key change weight changes the weight value for an entry in the pq and then calls bubble up or percolate down as necessary to readjust has key returns true or false for whether this key is in the PQ can also be implemented as find key how would you implement them?

13 Correctness we will not go into the proof at this point (although it s not a hard proof) Prim s algorithm is guaranteed to find a MST provided all edge weights are positive MST is not necessarily unique

14 Correctness we will not go into the proof at this point (although it s not a hard proof) Prim s algorithm is guaranteed to find a MST provided all edge weights are positive MST is not necessarily unique Prim s is a greedy algorithm what data structure is integral to Prim s greedy rule?

15 Edsger Dijkstra , Netherlands, a giant of CS

16 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming

17 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming argued for structured programming: procedures, loops, if-then-else branching, and against gotos

18 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming argued for structured programming: procedures, loops, if-then-else branching, and against gotos helped design and wrote first compiler for Algol Algol Pascal Ada

19 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming argued for structured programming: procedures, loops, if-then-else branching, and against gotos helped design and wrote first compiler for Algol Algol Pascal Ada invented the run-time stack to allow recursive procedures

20 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming argued for structured programming: procedures, loops, if-then-else branching, and against gotos helped design and wrote first compiler for Algol Algol Pascal Ada invented the run-time stack to allow recursive procedures invented parallel programming semaphores to prevent deadlock

21 Edsger Dijkstra , Netherlands, a giant of CS championed mathematical logic as the basis of computer programming argued for structured programming: procedures, loops, if-then-else branching, and against gotos helped design and wrote first compiler for Algol Algol Pascal Ada invented the run-time stack to allow recursive procedures invented parallel programming semaphores to prevent deadlock 1972 Turing Award

22 Shortest Paths Weighted in a weighted graph (directed or undirected), the path length is the sum of the weights along the path has nothing to do with the number of edges along the path what is the shortest path from A to D? 1 B 10 A 4 D 5 C 9 note that a naive greedy algorithm fails

23 Solution Algorithms many problems in CS are based on finding a solution is this a connected graph? what is the dfs search tree? what is the kth smallest element?

24 Optimization Algorithms many other problems in CS are based on optimizing a solution you make a 57 purchase and use a $1 bill to pay for it the amount of change is not in question: 43 but the optimal number of coins in the change is an issue you would be unhappy if the clerk gave you 43 pennies because you can form 43 with only 6 coins

25 Optimization Algorithms an optimization algorithm does not seek a solution it seeks the optimal solution almost always the minimum or maximum number of something for making change, it s the minimum number of coins

26 Optimal Solutions there is a simple strategy for solving every optimization problem formulate every possible combination of items (e.g., coin values) that give a solution choose the one combination that is optimal this strategy always finds the answer but what is the problem with this approach?

27 Optimal Solutions there is a simple strategy for solving every optimization problem formulate every possible combination of items (e.g., coin values) that give a solution choose the one combination that is optimal this strategy always finds the answer but what is the problem with this approach? the problem with this strategy is that all possible combinations is a factorial number of things

28 Greedy Algorithms in contrast, a greedy algorithm strategy works like this at each step, make a locally optimal choice hope that the accumulation of locally optimum choices is a globally optimum solution

29 Greedy Algorithm Problems greedy algorithms do not always yield globally optimal solutions problems that can be solved by greedy algorithms exhibit two characteristics 1. Greedy choice property: it is possible to make a choice that seems best by looking only at the current state and possibly previous states, but never requires a consideration of future states and never requires backing up to try something different. The choice is dictated by some ordering. 2. Optimal substructure property: an optimal global solution contains within it optimal solutions to subproblems.

30 Dijkstra s Algorithm Dijkstra invented many algorithms his most famous, named for him, is an algorithm for finding weighted shortest paths it is a greedy algorithm A 4 B C 1 D E F

31 Dijkstra s Algorithm void dijkstra(graph, start, parent) { parent.at(start) = start; pq.build_heap(start = 0, all other vertices ); for (i in 0.. n - 1) { vertex = pq.top(); pq.pop(); min_cost = vertex.value; foreach w adjacent to vertex { if (pq.has_key(w) && min_cost + graph_weight(vertex:w) < pq.key_value(w)) { parent.at(w) = vertex; pq.change_weight(w, min_cost + graph_weight(vertex:w)); } } } }