Dynamic Programming. Lecture #8 of Algorithms, Data structures and Complexity. Joost-Pieter Katoen Formal Methods and Tools Group
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1 Dynami Programming Leture #8 of Algorithms, Data strutures and Complexity Joost-Pieter Katoen Formal Methods and Tools Group Otober 29, 2002 JPK
2 #8: Dynami Programming ADC (214020) Overview What is dynami programming? Fibonai numbers revisited Subproblem graphs Dynami programming version of a reursive algorithm Multiplying a sequene of matries Optimal binary searh trees JPK1
3 #8: Dynami Programming ADC (214020) What is dynami programming? Main original motivation: [Bellman 1957] replae an exponential-time omputation by a polynomial-time omputation Splitting problems o sub-problems may be very expensive if not ontrolled orretly, many subproblems will be solved repeatedly Dynami programming: basi idea: store results for subproblems rather than reomputing them appliable to problems where a reursive algorithm solves many subproblems repeatedly JPK2
4 #8: Dynami Programming ADC (214020) Fibonai numbers revisited Consider the growth of a rabbit population, e.g.: suppose we have two rabbits, one of eah sex rabbits have bunnies one a month after they are 2 months old they always give birth to twins, one of eah sex they never die and never stop propagating The # rabbits after months is omputed by: for We thus obtain the sequene: JPK3
5 JPK4 fibre #8: Dynami Programming ADC (214020) A naive algorithm fibre if return else return fibre The # arithmeti steps needed to ompute fibre is: The omplexity of fibre is exponential: fibre for
6 #8: Dynami Programming ADC (214020) Subproblem graphs The subproblem graph of a reursive algorithm A for a problem is a direted graph verties are instanes (or: inputs) for this problem iff when A is invoked on it makes a diret reursive all to Subproblem graph A for problem instane of algorithm A is: the portion of the subproblem graph of A that is reahable from vertex If algorithm A always terminates then its subproblem graph is ayli JPK5
7 #8: Dynami Programming ADC (214020) Subproblem tree for Fibonai funtion vertex labels are the parameters of the reursive alls JPK6
8 #8: Dynami Programming ADC (214020) Subproblem graph for Fibonai funtion note that this is a dependeny graph JPK7
9 #8: Dynami Programming ADC (214020) Graph erpretation of dynami programming A natural reursive omputation making reursive alls as needed: is like a memoryless traversal of the subproblem graph of A memoryless traversal = DFS without oloring verties gray if there are and a DAG an have exponentially many paths! Consider the subproblem graph as a dependeny graph paths to a subproblem, this subproblem will be solved times if has edges to The essene of dynami programming is: these problems must be solved prior to solving to exeute the subproblems suh that eah needs to be solved only one and store the subproblem solutions for later use memo-ization JPK8
10 #8: Dynami Programming ADC (214020) Dynami programming version of reursive algorithm A dynami programming version DP(A) of reursive algorithm A: is a depth-first searh traversal of the subproblem graph A solutions found are stored in a ditionary ADT memo-ization Operations for ditionary : reate member retrieve store : reates an empty ditionary of items : heks whether item belongs to the ditionary : retrieves item requires ; to be in : store information item for Implementation of ditionary depends on the appliation JPK9
11 2. Prior to returning solution to store it (i.e., olor #8: Dynami Programming ADC (214020) Dynami programming version of reursive algorithm Convert algorithm A on problem o dynami version DP(A) by: 1. Prior to reursive all on subproblem hek if solution to is stored No? Do the reursive all ( is white), and add edge Yes? Retrieve the solution and perform no reursive all ( to the DFS tree is blak) blak) requires the subproblem graph to be ayli JPK10
12 if #8: Dynami Programming ADC (214020) Dynami programming version of Fibonai funtion fibdp Dit soln fib if fib else member soln // is fib already omputed? fibdp soln // reursive all else retrieve soln // retrieve stored solution if member soln // is fib already omputed? fibdp soln // reursive all else retrieve soln // retrieve stored solution fib store soln fib // store fib return fib worst-ase time omplexity is in ; spae omplexity JPK11
13 #8: Dynami Programming ADC (214020) Topologial sorting revisited A topologial order for DAG is: an assignment topo of dist egers suh that for every edge topo is alled the topologial number of we have topo A reverse topologial order for DAG to the verties of topo is as above but for every edge we have topo topo How to find a (reverse) topologial order: use DFS or analyse problem Appliation of reverse topologial order to dynami programming: a shedule of subproblems an be found suh that on solving a subproblem it is guaranteed that all subproblems on whih it depends have been solved thus: always look-up, no hek neessary JPK12
14 #8: Dynami Programming ADC (214020) Topologial sorting revisited Nr Task Depends on A hoose lothes I B dress A, H C eat breakfast E, F, G D leave B, C E make offee I F make toast I G pour juie I preedene graph transpose of depends on H shower I I wake up does there exist a shedule for this problem? JPK13
15 #8: Dynami Programming ADC (214020) Topologial sorting revisited dependeny graph drawn aording to the reverse topologial order means that task needs to be ompleted prior to task JPK14
16 #8: Dynami Programming ADC (214020) Simplifiations for Fibonai-DP There is a straightforward reverse topologial order for Fibonai: namely just inreasing order of parameter tasks an be sheduled suh that their results are always available when needed Any subproblem only depends on its two predeessor subproblems not neessary to store all results, but last two results suffies JPK15
17 #8: Dynami Programming ADC (214020) Simplified Fibonai-DP fibiter urr for urr ppred pred predppred ppred pred pred urr return urr // shift ppred // shift pred The time omplexity of fibiter is linear: fibiter The spae omplexity is JPK16
18 #8: Dynami Programming ADC (214020) Overview What is dynami programming? Fibonai numbers revisited Subproblem graphs Dynami programming version of a reursive algorithm Multiplying a sequene of matries Optimal binary searh trees JPK17
19 JPK18 #8: Dynami Programming ADC (214020) Matrix multipliation Reall matrix multipliation: element- Multiplying a -matrix and a wise multipliations -matrix requires Matrix multipliation is assoiative:
20 Mults JPK19 #8: Dynami Programming ADC (214020) The matrix multipliation order problem What is the best order to ompute for? where matrix Consider matries has ardinality,,, Multipliation order 1 produt produt 3 + produt = 20,700 = 11,750 = 41,200 = 1,400 What is the minimal number of multipliations? What is the order?
21 #8: Dynami Programming ADC (214020) How to parenthesize the produt of matries? Let be the number of different parenthesizations of matries Reursive equation for the number of parenthesizations matries matries is: if if there are exponentially many possible parenthesizations JPK20
22 #8: Dynami Programming ADC (214020) A naive solution Let the first matrix multipliation be at position : Reursively solve the remaining subproblem: Do this for any and selet the one with minimal ost Reurrene relation: show this Subproblem graph has exponentially many verties every subsequene of with elements is a reahable subproblem JPK21
23 #8: Dynami Programming ADC (214020) A naive solution matmult1 bestcost if else dim, seq bestcost bestcost for // outline // one matrix or none ost of multipliation at position seq newseq matmult1 seq with th element deleted dim, newseq bestcost min bestcost ; return bestcost JPK22
24 as #8: Dynami Programming ADC (214020) Alternative problem deomposition Let the last multipliation be at position The two remaining subproblems are: (of dimension (of dimension last step is to multiply and ) ) Identify original problem ) after hoie of, the remaining subproblems are and This reates verties in the subproblem graph To determine the minimal ost we need to hek all possibilities for JPK23
25 JPK24 dim #8: Dynami Programming ADC (214020) Alternative problem deomposition matmult2 bestcost if else bestcost for matmult2 matmult2 dim bestcost min dim dim bestcost dim dim bestcost // solve subproblem // one matrix or none ; // solve subproblem ; // solve subproblem // ost of multipliation at position return bestcost, i.e.,
26 The subproblem graph of #8: Dynami Programming ADC (214020) Subproblem graph example, i.e., subproblem : (0,4) (0,1) (1,4) (0,2) (2,4) (0,3) (3,4) (1,2) (1,3) (2,3) JPK25
27 #8: Dynami Programming ADC (214020) Subproblem graph analysis Verties are identified by with this yields For vertex verties in total two subproblems have to solved: and for allwith thus, the number of edges per vertex is less than Subproblem graph has verties and at most edges worst-ase time-omplexity of a DFS on the subproblem graph is It is feasible to onvert reursive problem o DP version as this boils down to a DFS on the subproblem graph this redues the time-omplexity from to JPK26
28 ost #8: Dynami Programming ADC (214020) Dynami programming version matmult2dp dim bestcost if bestcost else bestcost for if member ost matmult2dp dim ost ; else retrieve ost // look-up solution if member ost matmult2dp dim ost ; else retrieve ost // look-up solution dim dim dim // ost of multipliation at position bestcost min bestcost store ost bestcost ; // store obtained result return bestcost JPK27
29 #8: Dynami Programming ADC (214020) The dependeny graph of Reverse topologial order, i.e., subproblem : (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) ost if min ost ost if JPK28
30 #8: Dynami Programming ADC (214020) Simplified DP matrix-multipliation order matmultorder ost for for if dim bestcost for ost ost ost dim bestcost dim dim bestcost min bestcost bestcost else bestcost // iterate over the rows // iterate over the olumns // onsider // look-up solution // look-up solution // ost of multipliation at position // store obtained result return ost JPK29
31 #8: Dynami Programming ADC (214020) Example omputation ost if min ost ost if JPK30
32 #8: Dynami Programming ADC (214020) Overview What is dynami programming? Fibonai numbers revisited Subproblem graphs Dynami programming version of a reursive algorithm Multiplying a sequene of matries Optimal binary searh trees JPK31
33 #8: Dynami Programming ADC (214020) Binary searh trees revisited In a binary searh tree (BST) keys are stored suh that: the key at a node is at least all keys in its left subtree the key at a node is at most all keys in its right subtree An inorder traversal of a binary searh tree yields a sorted list of keys two binary searh trees ontaining 2, 3, 5, 6, 7, 9 5 JPK32
34 suh that #8: Dynami Programming ADC (214020) Optimal binary searh trees Keys with probability to be sought Average nodes examined in BST with keys is: where is the number of omparisons made to loate key is the weighted retrieve ost for key in BST How to organise is minimal? how to organise the key values in a BST to minimise the average number of key omparisons? JPK33
35 #8: Dynami Programming ADC (214020) Optimal BSTs Let be the minimum weighted ost for keys if is hosen as root where is the probability to searh some key in This an be simplified to: The problem is now to ompute min Naively, a reursive algorithm yields an exponential time-omplexity JPK34
36 #8: Dynami Programming ADC (214020) DP solution to optimal BSTs float optimalbst prob float ost float bestcost for // iterate over the rows for // iterate over the olumns if bestcost else bestcost for ost ost ost bestcost min bestcost bestcost // store obtained result return ost JPK35
37 #8: Dynami Programming ADC (214020) Dynami programming reipe 1. Takle the problem top-down ; this yields a reursive algorithm 2. Define an appropriate ditionary; transform the reursive solution o a DP algorithm 3. Complexity of DP-algorithm is omplexity DFS on subproblem graph 4. Choose an appropriate data struture for implementing the ditionary 5. If possible, analyse the subproblem graph and find a reverse topologial order; simplify your DP-algorithm aordingly 6. Deide how to get the solution to the problem from the data in the ditionary JPK36
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