Automation of the Solution of Kakuro Puzzles

Transcription

1 Automation of the Soution of Kakuro Puzzes R. P. Davies, P. A. Roach, S. Perkins Department of Computing and Mathematica Sciences, University of Gamorgan, Pontypridd, CF37 1DL, United Kingdom, Abstract Kakuro puzzes, aso caed cross sum puzzes, are grids containing cues to the competion of numerica words. Being structured in a simiar way to crossword puzzes, Kakuro grids contain overapping continuous runs that are excusivey either horizonta or vertica. The cues take the form of specified run totas, and a puzze is soved by pacing a vaue in each ce such that every run sums to its specified tota, and no run contains dupicate vaues. Whie most puzzes have ony a singe soution, onger runs may be satisfied using many arrangements of vaues, eading to the puzze having a deceptivey arge search space. The associated, popuar Sudoku puzze has been inked with important rea-word appications incuding timetabing and confict free waveength routing, and more recenty, coding theory due to its potentia usefuness in the construction of erasure correction codes. It is possibe that Kakuro puzzes wi have simiar appications, particuary in the construction of codes, where run totas may form a generaised type of parity check. A project has begun to investigate the properties of the Kakuro puzzes, and thereby estabish its potentia usefuness to rea-word appications incuding coding theory. This paper reports some eary findings from that project, specificay concerning puzze compexity and the appropriateness of heuristic approaches for its automated soution. It highights the use of heuristics to guide search by a backtracking sover, in preference to oca search optimisation, and reports on the effectiveness of two heuristics and a pruning technique for reducing soution time. The authors beieve this to be the first pubished work in the use of heuristics, in combination with pruning, for the automated soution of Kakuro puzzes. 1 Introduction Kakuro puzzes are number puzzes that have strong simiarities with the more famiiar crossword puzze, due to their use of cues to specify correct numerica words within a grid structure. Unike crosswords, Kakuro puzzes more easiy transcend anguage barriers due to their use of number sequences. Puzzes of this

2 R. P Davies, P. A. Roach, S. Perkins type typicay consist of an n m grid containing back and white ces. A white ces are initiay empty and are organised into overapping continuous runs that are excusivey either horizonta or vertica. A run-tota, given in a back cue ce, is associated with each and every run. The puzze is soved by entering vaues (typicay in the range 1,..., 9 incusive) into the white ces such that each run sums to the specified run-tota and such that no vaue is repeated in any horizonta or vertica run. Most pubished puzzes consist of an n m grid and are we-formed [6], meaning that ony one unique soution exists. Such puzzes are aso caed promiseprobems (the promise being a unique soution) [1]. The puzzes are designed so that this unique soution may be determined through the empoyment of a range of types of ogica deduction and reasoning; no guesswork shoud be needed. Many puzzes have refective or rotationa symmetry, athough this is ony to improve the visua appearance of the grid. The name Kakuro comes from the Japanese pronunciation of the Engish word cross appended to the Japanese word for addition. The name was part of a rebranding by Japan s Nikoi Puzzes Group of De Magazines Cross Sum puzzes, which can be traced back as eary as 1966 [4]. Currenty, the popuarity of Kakuro in Japan is reported second ony to Sudoku puzzes [4], but it is ony during the ast four years that the puzze has gained wider goba popuarity, particuary in the West. Reated puzzes incude: Cryptic Kakuro [11], in which aphametric cues must be soved as a prerequisite to the Kakuro puzze itsef; Cross-sum Sudoku [11], which combines the rues of standard Kakuro puzzes with the constraints of standard Sudoku puzzes; Cross Products, in which cue ces suggest the product of digits in a run, rather than their sum; and Survo Puzzes [7], in which the vaues 1,..., mn must be paced, once each, into an n m grid that often contains givens, so as to satisfy row and coumn sums. The associated, popuar Sudoku puzze has been inked with important reaword appications incuding timetabing [5] and confict free waveength routing [3], and more recenty, coding theory due to its potentia usefuness in the construction of erasure correction codes [10]. At present, very itte has been pubished specificay on Kakuro and its reated puzzes. The authors have previousy reported on the use of binary integer programming and oca search approaches to the soution of Kakuro [2], concuding on the need for heuristics to guide soution and for the need to reduce the size of the search space to be examined. The soution of Kakuro puzzes has been shown to be NP-Compete [9], through demonstrating the reationship between Kakuro and the Hamitonian Path Probem, and 3SAT (the restriction of the Booean satisfiabiity probem). It is possibe that Kakuro-type puzzes wi have simiar appications to Sudoku, particuary in the construction of codes, where run totas may form a generaised type of parity check. A project has begun to investigate the properties of the cass of Kakuro puzzes, and thereby estabish its potentia usefuness within a range of appications, in-

3 Automation of the Soution of Kakuro Puzzes cuding coding theory. This paper reports some eary findings from that project, specificay concerning puzze compexity and the appropriateness of heuristic approaches for its automated soution. It highights the use of heuristics to guide search by a backtracking sover, in preference to oca search optimisation. Evauation is presented of the effectiveness of two heuristics for guiding search and a method for pruning the search space that need be considered by the sover. The authors beieve this to be the first pubished work in the use of heuristics, in combination with pruning, for the automated soution of Kakuro puzzes. 2 Probem Anaysis Let a Kakuro grid be termed K, where K has dimension n m, and the ce at row i and coumn j is termed k i, j. Each ce is either a white ce (to be assigned a numerica vaue in the range 1,..., 9 ) or a back cue ce. Grid K contains a coection of runs of white ces. Each of these individua runs is excusivey either horizonta or vertica and is termed a tupe r ( = 1,..., p ) where r r, the set of a tupes, and p is the number of runs contained in the puzze grid. We define the tupe to be such that it contains no repeated eements. r Therefore r is described either as a tupe of connected horizonta white ces or of connected vertica white ces. A horizonta run is defined: r = ( k i j,..., k i, j ) ki j x r, s x e,, s e where the run is in row i ( 1 i n ), beginning in coumn j s and ending in coumn j e, ( 1 j < j m). A vertica run is defined: s e r = ( k i, j,..., k i, j ) k, j r, s x e s e i x where the run is in coumn j ( 1 j m ), beginning in row i s and ending in row i e ( 1 i < i n ). s e The vaues to be paced within each white ce, constraints, namey that: In each and every run, ce: k i, j u k i,j, are governed by puzze r r, the same vaue must appear in no more than one k i, j v ki j x r,, ju jv

4 R. P Davies, P. A. Roach, S. Perkins k i u, j k i v, j k j r In each and every run, fied: k i, j r k i = t, j i x,, iu i. v r r, the corresponding run-tota, t, must be satis- Kakuro grids can vary in the difficuty of their soution. Generay, the comp exity of a given puzze cannot be determined by the size of the grid aone, but shoud instead be determined from a combination of the number of empty (white) ces in its initia state, the grid size and the magnitudes of the run-totas. In order to estabish the potentia size of the search space for a puzze, we coud consider the number of options for assigning vaues to a ce and thereby determine an upper bound for the number of different grid arrangements of vaues. (We note that, for a we-formed puzze, a but one of these grids, the unique soution, woud be invaid due to the puzze constraints.) Fig. 1. A 5 5 initia puzze grid A crude upper bound for the number of grid arrangements for a puzze grid with w w white squares is 9, since each square can take any of nine numerica vaues, assuming the standard range of vaues ( 1,..., 9 ) is being used. The puzze grid in Fig. 1, with sixteen white squares, woud therefore have an upper bound of x possibe arrangements. This upper bound is greaty reduced by considering which of the nine avaiabe vaues can egitimatey be paced in each of the white ces, c i, depending on the run(s), r (each with corresponding run-tota t ), in which the ce resides.

5 Automation of the Soution of Kakuro Puzzes A set of vaues, P, that may be assigned to ces in a run r is constructed, such that: P = {1,, 9} if > 9 P = {1,, a-1} if t = a, a 9 The improved upper bound woud then be: w i= 1 min{ P c i r } t For exampe, the white ce at the uppermost eft of the grid in Fig. 1 is a member of a run totaing 11 and of another totaing 5. Concentrating on the ower of the two run-totas, ony a vaue in the range 1,..., 4 can be paced in this ce. When a ces are considered in this way, a new upper bound equaing 1,133,740,800 arrangements can be cacuated for this exampe. The positioning of runs, and the seections of run totas of Kakuro puzzes can vary greaty. This makes the task of devising a genera formua for the exact number of possibe Kakuro grid arrangements of a given size difficut, if not impossibe, to achieve. Instead we focus on determining the tota number of arrangements of vaues within a singe run which woud satisfy the puzze constraints the run tota constraint, and the requirement to have no dupicated vaues in the run. Different sets of distinct vaues might meet the run constraints, but each set can be permuted into different orderings ony one of which wi match the puzze soution. The tota number of such arrangements of vaues can be derived from the coefficients obtained from a series expansion of the generating function: F ( x) = r! 9 i= 1 i (1 + x a) r x t The coefficient of a represents the number of ordered ce vaue compositions of ength r that have no repeated vaue and that sum to t. This generating function has been used to deveop a ook-up tabe that is empoyed in a heuristic in Sect It is worth noting that the function is generaisabe to Kakuro puzzes that use arger sets of assigned vaues (i.e. beyond 1,..., 9 ). 3 Automating the Soution Automated approaches to the soution of a given Kakuro puzze can be paced into two categories. One category of approaches woud use simiar methods to those used by a human sover, where the constraints of the puzze (run-totas and non-

6 R. P Davies, P. A. Roach, S. Perkins dupication of vaues within runs) are considered in turn in some ogica order unti a vaid soution is found. Aternativey, the secondary category woud use search agorithms, possiby aong with heuristics and objective functions for optimisation. These heuristics and objective functions woud incorporate probem domain information. 3.1 Seecting a Suitabe Approach A given Kakuro puzze coud be soved exhaustivey. That is, a possibe vaues are tried in a ces, fuy enumerating the search space in order to ocate the soution. This approach is adequate for smaer grids or when a smaer range of numbers is to be used but very time consuming and inefficient for most puzze grids where very arge numbers of puzze states woud have to be checked. The puzze constraints, non-dupication of vaues within runs and the summation requirement of vaues to a specified run-tota, make the puzze seemingy appropriate for a constraint-based approach to a soution. Binary integer programming is one such approach, and such a formuation of the puzze has previousy been presented by the authors [2]. In that formuation, ten binary decision variabes,, are associated with every ce, where row i and coumn j specify the A i,j,k ce position, and k specifies an avaiabe vaue for assignment to the ce (with zero indicating a back square). Puzze constraints and trivia constraints (such as there ony being one vaue per ce and ony vaues in the range 1,..., 9 can be added to white ces) are expressed expicity. The soution is indicated by the coection of binary decision variabes that are set to 1, showing which vaue k shoud be assigned to the ce at row i and coumn j. The resuts for this approach, using XPress MP (a suite of optimisation packages) showed that this approach works we for sma puzzes [2]. However, the arge number of binary decision variabes for arger grids may make this an inefficient genera approach. The difficuty of search space size might be overcome by the use of heuristics in a oca search optimisation approach [8]. This approach woud empoy some objective function to determine an efficient path through the search space, beginning from some initia state. This initia state might be a particuar arrangement of vaues within ces such that run dupication constraints are met but not run-tota constraints. An operator woud then change the vaues within one or more of the ces so that successor states, different arrangements of vaues within ces, are produced foowing each iteration of the search (as iustrated in Fig. 2).

7 Automation of the Soution of Kakuro Puzzes Fig. 2. A sampe search space containing puzze states A states woud then be evauated and scored by the objective function, so that the state with highest score woud be expored next. However, such an approach is ony feasibe if a sensibe and effective objective function can be constructed, such that it is possibe to move reiaby towards the goa state. Unfortunatey, the amount of probem domain information that can usefuy be incorporated into an effective objective function is imited. This puzze information reates specificay to how cosey the current horizonta and vertica run sums match the specified run-totas. There is a ikeihood of many different states mapping to the same score and of the method becoming stuck in pateaus [8] in the search space. Simiar difficuties have been reported in a oca search optimisation approach to the soution to Sudoku puzzes [6]. Aso, since each vaue in a particuar ce can be repaced by up to eight aternative vaues, the search space can grow very rapidy. Soutions of arger puzzes woud inevitaby invove the storage of a very arge number of states. Meta-heuristic approaches might be empoyed to overcome the imitations of the objective function, however the authors wish to empoy probem domain information more directy. For the above reasons, search optimisation approaches are not pursued here. Instead we empoy a backtracking approach to soving Kakuro puzzes, as described beow. 3.2 Backtracking Sover An approach that takes more direct advantage of the probem compexity characteristics notaby the permutations of the vaues that may egitimatey be assigned to runs is desired. A backtracking agorithm, empoying a depth-first approach to examining the search space, can be made appropriate for the soution of Kakuro puzzes if suitabe heuristics to guide the backtracker, and effective pruning conditions can be determined to reduce search space size. In this section, a backtracking agorithm, impemented through the use of a stack, is described that incorporates

8 R. P Davies, P. A. Roach, S. Perkins conditions to prune parts of the search space in which vaid soutions wi definitey not be ocated. BACKTRACKING ALGORITHM Initiaise puzze information and stack. Current_State becomes the initia-state. Add Current_State to stack. Current_Ce is set to be the first avaiabe white ce. Current_Vaue = 1. WHILE [empty white ces exist] Pace Current_Vaue into Current_Ce. Increment Iteration_Count. Determine runs in which Current_Ce resides, and corresponding run totas. IF [no dupicates in runs] and ([run-tota(s) not exceeded] or [run(s) competed correcty]) Push Current_State to stack. IF [empty white ces exist] Current_Ce becomes next avaiabe ce. END-IF Reset Current_Vaue to 1. ELSE-IF ([runs under target run-totas] or [dupicate in run(s)]) and [Current_Vaue<9] Current_Vaue = Current_Vaue +1. ELSE Pop state from stack to become Current_State. Current_Ce becomes previous ce. Current_Vaue becomes vaue within Current_Ce. WHILE [Current_Vaue = 9] Pop state from stack to become Current_State. Current_Ce becomes previous ce. Current_Vaue becomes vaue within Current_Ce. END-WHILE Current_Vaue = Current_Vaue +1. END-IF END-WHILE Output Current_State as soution. This approach begins with an empty grid and attempts assignments of vaues to each white ce in turn, starting with the owest numerica vaue, and beginning the pacements from the top eftmost ce. It foows a depth-first [8] enumeration of the search space, favouring the assignment of ow numerica vaues, but tests within the agorithm ensure that some fruitess paths through the search space are avoided. An apparenty successfu assignment of a vaue to a ce (one which does not vioate puzze constraints) resuts in the current grid being pushed onto the stack. Vioations of the puzze constraints a dupicate vaue in a run, or an ex-

9 Automation of the Soution of Kakuro Puzzes ceeded or under-target run tota where a possibe vaues have been considered for the fina ce of a run resut in the agorithm backtracking, and popping the ast successfu grid state from the stack. The stack ony stores incompete states, that are apparenty vaid, aong one branch of the search space, thus avoiding the memory based issues which can arise in search approaches in which a vaid partia states encountered are stored (for exampe in the queue of a oca search optimisation approach [8]). An iteration count is incremented each time an attempt is made to assign a vaue to a ce, and is used as a measure of agorithm performance in Sect. 4. Whie this approach is idea for smaer puzzes, the agorithm can be required to perform a great dea of backtracking in arger puzzes. This necessitates the addition of further components. The heuristics and pruning conditions that have been tested in this project are described in Sect. 3.3 beow. 3.3 Modifications to the Backtracking Agorithm In this section, three modifications to the Backtracking Agorithm of Sect. 3.2 are proposed. The resuts of using these approaches are presented and anaysed in Sect Ce Ordering It is proposed here that the path taken through the search space be guided by consideration of how many vaid arrangements of vaues there are for each run. The ce ordering heuristic empoyed is that by favouring the competion of ces in runs having fewest vaid arrangements, a reduction can be achieved in the maximum amount of backtracking required due to incorrect assignments to ces considered near the start of the search process. Those ces in runs having most potentia vaid arrangements wi be considered ater, tending to push the consideration of ces requiring most backtracking to a deeper eve in the search space. As an exampe, a run-tota of 6 over two ces can be fied using the tupes (1, 5), (5, 1), (2, 4) and (4, 2). (The tupe (3, 3) woud be invaid due to the dupication constraint). Hence this run can be fied in four different ways. A ook-up tabe is constructed using the generating function of Sect. 2. This tabe expicity states how many distinct compositions of vaues exist for each runtota t and a possibe run engths r. As this approach uses cacuations based on entire runs, rather than singe ces, a ce inherits the owest number of choices of any run in which it is situated. This represents an upper bound for the actua number of choices for that ce. (We note that a more accurate measure is to be found in the intersection of the arrangements in runs, which is more difficut to cacuate, and remains as future work.)

10 R. P Davies, P. A. Roach, S. Perkins Reverse Vaue Ordering This heuristic favours the assignment of vaues in the range 1,..., 9 in reverse order, essentiay being based on the assumption that puzzes wi be soved more quicky in this manner. Ceary, a vaues are equay ikey to be the content of a ce of a puzze soution, in a genera sense; the actua ikeihood of, for exampe, a 1 or 9 appearing more frequenty in a soution wi be puzze-specific. This is a poor heuristic, but no worse in genera that the reverse assumption. Hence it provides a usefu test of the performance of the agorithm, when measuring the resuts of many puzzes. A puzze having severa high vaues in ces considered at the start of soution wi probaby sove more quicky when using this heuristic Projected Run Pruning The Backtracking Agorithm of Sect. 3.2 checks for invaid assignments to a run on the competion of that run. This wi sti aow poor choices of vaues to be paced at the beginning of a run, such that the run tota can not be met with egitimate vaue assignments in the remaining ces. As an exampe, consider a run of 5 ces having the run tota 35. A pacement of 1 in the initia ce wi seem egitimate, but even the assignment of the argest vaues to the remaining ces 9, 8, 7 and 6 wi ony ead to a tota of 31. In such a case, considerabe processing time woud be wasted attempting to fi the remaining ces, unti the Backtracking Agorithm eventuay paces a vaue arger than 4 in the initia ce. By considering whether a run can possiby be competed to meet its tota, each time an assignment is made, fruitess branches of the search space can be pruned. An additiona vaidity check is added to the Backtracking Agorithm of Sect On assigning a vaue to a ce in a run that sti possesses unassigned ces, a cacuation is performed of the sum of the argest possibe vaues that may sti egitimatey be added to the remaining ces of that run. If this sum woud yied a run tota at east matching the specified run tota for that ce, the backtracker continues, otherwise this branch of the search space is pruned. This approach wi reduce the number of puzze states that need to be considered and hence shoud, in genera, decrease the time taken to obtain a soution to a given puzze. 4 Resuts and Timings There is no pubished work with which to compare the findings of this project, and so the resuts obtained using the heuristics and the projected run pruning from Sect. 3 wi be compared to resuts obtained using the backtracking sover aone, for specific puzzes of varying sizes. Tests were performed on a Vigen Inte Core

11 Automation of the Soution of Kakuro Puzzes 2 Duo processor 2.66GHz, with 2GB RAM. Programs were deveoped in Java (using Orace Jdeveoper ) and executed in the J2SE runtime environment. Initia experimentation focused on estabishing the reative and genera effectiveness of the methods proposed in Sect. 3, and resuts are shown in Tabe 1. Few puzzes of sma size were avaiabe for testing, but those tested were deemed sufficient to examine the methods and to demonstrate the puzze-specific nature of their effectiveness. The numbers of iterations (expained in Sec. 3.2) are shown for a range of puzze sizes. Tabe 1. Iteration counts for specific puzzes, in each method Puzze Grid Size Heuristic Used Projected Projected Backtracking Ce Vaue Run Pruning & Ce Run Aone Ordering Ordering Pruning Ordering (a) (b) 2, ,562 2, (c) (d) 2, , ,393,677 1,052, ,945 12,455,461 24, ,347 71,168 3,140 7,032 27,440 As woud be expected, the reverse vaue ordering worked best on certain puzzes these being ones in which the first few ces that considered had high vaues. Ce ordering was often effective, but seemed ess so for arger puzzes for certain puzzes it performed worse than backtracking aone. In contrast, the projected run pruning performed more consistenty, never requiring more iterations than the backtracker aone (as woud be expected), and often requiring far fewer iterations. A method that encourages rapid and eary pruning is desired. The combination of ce ordering and projected run pruning occasionay reduced the number iterations beow the count achieved by either approach individuay, suggesting that the methods might combine we in guiding the search method to earier pruning of the search space. However, this behaviour was not consistent. Puzzes of sma size generay sove quite rapidy, but the processing overhead of the methods is of interest here. Tabe 2 shows the average time taken per iteration, measured in miiseconds, for the puzze set of Tabe 1, this time banded accord-

12 R. P Davies, P. A. Roach, S. Perkins ing to puzze size. As woud be expected, the average time per iteration is generay higher for smaer puzzes, as the search spaces are sma, hence the benefits of pruning are ess significant. It seems reasonaby cear that the processing overheads of ce ordering (arising from a pre-processing step and indexing of an array) and pruning are sma. Tabe 2. Average time (miiseconds) taken per iteration Heuristic Used Backtracking Aone Ce Ordering Vaue Ordering Project Run Pruning Projected Run Pruning & Ce Ordering Puzze Grid Size < 5x x5 & 6x > 6x Whie few puzzes of sma sizes are avaiabe, a arger number of pubished puzzes exist for a more standard chaenge. For a test set of puzzes of size 9 9, we pursue the most promising methods of projected run pruning and its combination with ce ordering. Tabe 3 shows resuts for thirteen puzzes of grid size 9 9. Tabe 3. Iteration statistics for thirteen puzzes with grid size 9x9. Projected Run Pruning Projected Run Pruning & Ce Ordering Minimum Iterations Maximum Iterations Median Iterations Average Iterations % of cases where method performed best 5,829 1,554,208 65, , % 2,543 28,039, ,512 5,795, % In a sma number of cases, the combination of the ce-ordering heuristic and pruning improved resuts (shown by the minimum number of iterations and the percentage of cases where improvement occurred), but the median and maximum number of iterations show both that the combination is an unreiabe approach and that on certain puzzes, performance is greaty worsened. Hence the projected run pruning method is considered here to be the most reiabe approach.

13 Automation of the Soution of Kakuro Puzzes An extended test set of 20 puzzes of size 9 9 were soved using just projected run pruning. The fastest soution time (in miiseconds) was 21,096, the ongest 36,075,603, the median 323,694 and the average 4,675,317. The average time per iteration was ms to 4 decima paces. This approach is reativey promising, but further pruning methods to force an earier and more rapid reduction in search space size, and heuristics to guide search, are sought to enabe more rapid soution. 5 Concusion This paper has anaysed the size of a Kakuro search space. This incudes estabishing an improved upper bound for the number of possibe arrangements of vaues in a Kakuro grid. More significanty, a generating function has been presented to determine the exact number of vaid arrangements of vaues in any given run; this function can be used for different grid sizes and different numbers of vaues to be assigned. The suitabiity of a range of search approaches for the soution of Kakuro has been considered, and a backtracking approach has been presented as the preferred approach. A ce ordering heuristic, based on the number of vaid arrangements of vaues in a given run, has been proposed and evauated. Lasty, a pruning method has been proposed to reduce the part of the search space that need be examined, by checking whether a run tota can possiby be met each time an assignment is made to a ce in that run. The pruning method proved to be most effective in reducing soution time for a range of puzze grids. The ce ordering heuristic performed unreiaby, making reasonabe improvements in the soution time in some cases, but greaty increased soution time in other cases. This heuristic might be improved by estabishing the intersection of the arrangements in runs, rather than aowing a ce to inherit the owest number of possibe arrangements of vaid soutions of the two runs in which it is situated, thus giving a better measure of possibiities for a singe ce. The competion of a started run, in preference to continuay jumping to the next singe ce with fewest apparent choices, might aso aow the earier detection of fruitess branches. The usefuness of Kakuro for appications, incuding Coding Theory, wi depend in part on the deveopment of methods to reiaby enumerate the search spaces of specific puzzes more rapidy. A more detaied understanding of the size of the search spaces of puzzes wi aso be required. For this, it is proposed that the resuts of this paper be extended through further improvements to the upper bound for the number of possibe arrangements of vaues in soution grids, through consideration of the intersection of runs. Acknowedgments The authors wish to thank Sian K. Jones for many hepfu discussions reating to this work.

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