Other partial solution strategies

Partial solution algorithms greedy  branch and bound  A*  divide and conquer  dynamic programming

4 queens: separate row and column complete solutions possible pruning More in next slide set…

Branch and Bound  eliminate subtrees of possible solutions based on evaluation of partial solutions complete solutions

e.g., branch and bound TSP complete solutions current best solution distance 498 distance so far 519

Branch and Bound requirement  a bound on best solution possible from current partial solution (e.g., if distance so far is 519, total distance is >519)  ‘tight’ as possible  quick to calculate  a current best solution (e.g., 498)

Tight bounds  value of partial solution plus  estimate of value of remaining steps must be ‘optimistic estimate’ example: path so far:519 remainder of path: 0 (not tight) bounded estimate:519

Optimistic but Tight Bound optimistic tight Current best Actual values if calculated

Example - b&b for TSP  Estimate a lower bound for the path length: Fast to calculate Easy to revise ABCDE A B C D87910 E

Example - b&b for TSP 1.Assume shortest edge to each city: ( ) = 40 A B C D E 2.Assume two shortest edges to each: ((7+8)+(7+7)+(9+10)+(7+8)+(10+11))/2 = = 42 Which is better? ABCDE A B C D87910 E

Tight Bound Path so far 0 40 Path so far Path so far = ( ) = 40 ABCDE A B C D87910 E Best Path found 46

B&B algorithm  Depth first traversal of partial solution space - leaves are complete solutions  Subtrees are pruned below a partial solution that cannot be better than the current best solution

Partial solution algorithms greedy branch and bound  A*  divide and conquer  dynamic programming

A* algorithm - improved b&b  Ultimate partial solution search  Based on tree searching algorithms you already know - bfs, dfs  Two versions: Simple - used on trees Advanced - used on general graphs

general search algorithm for trees* algorithm search (startState, fitnessFn()) returns bestState openList = new StateCollection(); openList.insert(startState) bestState = null bestFitness = min // assuming maximum wanted while (notEmpty(openList) && resourcesAvailable) state = openList.get() fitness = fitnessFn(state) if (state is complete and fitness > bestFitness ) bestFitness = fitness bestState = state for all values k in domain of next variable nextState = state.include(k) openList.insert(nextState) return solutionState *For graphs, cycles are a problem

Versions of general search  Based on the openList collection class Breadth first search Depth first search (->branch and bound) Best first search (informed search)  Best first  A*

A CBD FEGHIJKLMNOP QR ST UVWXYZabcdefghij kl mn algorithm search (startState, fitnessFn()) returns bestState openList = new StateCollection(); openList.insert(startState) bestState = null bestFitness = min // assuming maximum wanted while (notEmpty(openList) && resourcesAvailable) state = openList.get() fitness = fitnessFn(state) if (state is complete and fitness > bestFitness ) bestFitness = fitness bestState = state for all values k in domain of next variable nextState = state.include(k) openList.insert(nextState) return solutionState

A*  Best first search with a heuristic fitness function: Admissible (never pessimistic) Tradeoff: Simplicity vs accuracy/tightness  The heuristic heuristic: “Reduced problem” strategy e.g., min path length in TSP