SEMINAR 13:

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• SEMINAR 13
• Applications of backtracking

2
Problem 1

• Maps coloring Let us consider a geographical
  map containing n countries. Propose a coloring of
  the map by using mgt4 colors such that any two
  neighboring countries have different colors

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Problem 1

• Problem formalization Let us consider that the
  neighborhood relation between countries is
  represented as a matrix N of n rows and n columns
  as follows
• 0 if i and j
  are not neighbors
• N(i,j)
• 1 if i and j
  are neighbors
• Find a map coloring S(s1,,sn) with sk in
  1,,m denoting the color corresponding to
  country k such that for all pairs (i,j) with
  N(i,j)1 the elements si and sj are different
  (siltgtsj)

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Problem 1

• Solution representation
• S(s1,,sn) with sk representing the
  color associated to country k
• 2. Sets A1,,An 1,2,,m. Each set
  will be processed in the natural order of the
  elements (starting from 1 to m)
• Continuation conditions a partial solution
  (s1,s2,,sk) should satisfy siltgtsj for all pairs
  (i,j) with N(i,j)1
• For each k it suffices to check that
  skltgtsj for all pairs i in 1,2,,k-1 with
  N(i,k)1
• 4. Criterion to decide when a partial
  solution is a final one k n (all countries
  have been colored)

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Problem 1

• Recursive algorithm
• Coloring(k)
• IF kn1 THEN WRITE s1..n
• ELSE
• FOR j1,m DO
• skj
• IF validation(s1..k)
• THEN coloring(k1)

Validation algorithm Validation(s1..k) FOR
i1,k-1 DO IF Ni,j1 AND sisk THEN
RETURN False RETURN True
Call Coloring(1)
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Problem 2

• Path finding. Let us consider a set of n towns.
  There is a network of routes between these towns.
  Generate all routes which connect two given
  towns such that the route doesnt cross twice the
  same town

Routes from Arad to Constanta 1-gt7-gt3-gt4 1-gt2-gt3
-gt4 1-gt6-gt5-gt3-gt4 1-gt6-gt5-gt2-gt3-gt4
Towns 1.Arad 2.Brasov 3.Bucuresti 4.Constanta 5.
Iasi 6.Satu-Mare 7.Timisoara
Satu Mare
Iasi
Brasov
Arad
Timisoara
Constanta
Bucuresti
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Problem 2

• Problem formalization Let us consider that the
  connection information is stored in a matrix C
  as follows
• 0 if there doesnt exist a
  direct route between i and j
• C(i,j)
• 1 if there is a direct
  route between i and j
• Find all routes S(s1,,sm) with sk in 1,,n
  denoting the town visited at moment k such that
• s1 is the starting town
• sm is the destination town
• siltgtsj for all i ltgtj (a town is visited only
  once)
• C(si,si1)1 (there exists a direct route between
  towns visited at successive moments)

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Problem 2

• Solution representation
• S(s1,,sm) with sk representing the
  town visited at moment k
• 2. Sets A1,,An 1,2,,n. Each set
  will be processed in the natural order of the
  elements (starting from 1 to n)
• Continuation conditions a partial solution
  (s1,s2,,sk) should satisfy
• skltgtsj for all i in 1,2,,k-1
• C(sk-1,sk)1
• Criterion to decide when a partial solution is a
  final one
• sk destination town

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Problem 2

• Recursive algorithm
• routes(k)
• IF sk-1destination town THEN WRITE s1..k-1
• ELSE
• FOR j1,n DO
• skj
• IF validation(s1..k)
• THEN routes(k1)

Validation algorithm Validation(s1..k) IF
Csk-1,sk0 THEN RETURN False FOR i1,k-1
DO IF sisk THEN RETURN
False RETURN True
Call s1starting town routes(2)
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Problem 3

• Maze problem. Let us consider a maze defined on
  a nxn grid. Find a path in the maze which starts
  from the position (1,1) and finishes in (nxn)

Only white cells can be accessed. From a given
cell (i,j) one can pass in one of the following
neighboring positions
(i-1,j)
(i,j)
(i,j-1)
(i,j1)
(i1,j)
Remark cells on the border have
fewer neighbours
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Problem 3

• Problem formalization. The maze is stored as a
  nxn matrix
• 0 free cell
• M(i,j)
• 1 occupied cell
• Find a path S(s1,,sm) with sk in
  1,,nx1,,n denoting the indices
  corresponding to the cell visited at moment k
• s1 is the starting cell (1,1)
• sm is the destination cell (n,n)
• skltgtsqj for all k ltgtq (a cell is visited at
  most once)
• M(sk)0 (each visited cell is a free one)
• sk and sk1 are neighborhood cells

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Problem 3

• Solution representation
• S(s1,,sn) with sk representing the
  cell visited at moment k
• Sets A1,,An are subsets of 1,2,,nx1,2,,n
  . For each cell (i,j) there is a set of at most 4
  neighbors
• Continuation conditions a partial solution
  (s1,s2,,sk) should satisfy
• skltgtsq for all q in 1,2,,k-1
• M(sk)0
• sk-1 and sk are neighbours
• Criterion to decide when a partial solution is a
  final one
• sk (n,n)

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Problem 3
maze(k) IF sk-1(n,n) THEN WRITE s1..k ELSE
sk.isk-1.i-1 sk.jsk-1.j IF
validation(s1..k)THEN maze(k1)
sk.isk-1.i1 sk.jsk-1.j IF
validation(s1..k)THEN maze(k1)
sk.isk-1.i sk.jsk-1.j-1 IF
validation(s1..k)THEN maze(k1)
sk.isk-1.i-1 sk.jsk-1.j1 IF
validation(s1..k)THEN maze(k1)
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Problem 3
validation(s1..k) IF sk.ilt1 OR sk.igtn OR
sk.jlt1 OR sk.jgtn THEN RETURN False IF
Msk.i,sk.j1 THEN RETURN False FOR q1,k-1
DO IF sk.isq.i AND sk.jsq.j THEN
RETURN False RETURN True
Call of algorithm maze s1.i1 s1.j1 maze(2
)