Towers of Honoi

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Towers of Honoi

• An application of Recursion

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Towers of Hanoi Problem Invented by French
mathematician Lucas in 1880s. Original problem
set in India in a holy place called Benares.
There are 3 diamond needles fixed on a brass
plate. One needle contains 64 pure gold disks.
Largest resting on the brass plate , other disks
of decreasing diameters. Called tower of
Brahma. Priests are supposed to transfer the
disks from one needle to the other such that at
no time a disk of larger diameter should sit on a
disk of smaller diameter. Only one disk can be
moved at a time. Later setting shifted to
Honoi, but the puzzle and legend remain the
same. How much time would it take? Estimate..
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Finding a Recursive strategy Any tower with
more than one disk must be moved in pieces. If
there is just one disk, then move it. It must be
possible to break the problem into simpler
subproblems of same form.
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3 Disks
start
temp
finish
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Irrespective of number of disks, the following
steps need to be carried out The bottom most
disk needs to be moved to finish tower. So the
disks above it must be removed first in step
1. In step 2,the bottom most disk can be
moved. In step 3, the disks above it must be
replaced in proper position. Now let us consider
the problem of 3 disks.
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start
Step 1 Move 2 disks from start to temp using
finish Tower. To understand the recursive
routine, let us assume that we know how to solve
2 disk problem, and go for the next step.
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Step 2 Move the (remaining) single disk from
start to finish.
This does not involve recursion, and can be
carried out without using temp tower.
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Step 3 Now we are at the last step of the
recursive routine. Move the 2 disks from temp
tower to finish tower using the start tower (
recursively).
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This will solve the 3 disk problem.

Now let us see how the complete sequence can be
worked out and find out how many steps are going
to be actually needed.
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The Complete sequence for 3 Disks
start
temp
finish
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start
1
2
3
4
12
5
6
7
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void movetower ( n, start, finish, temp) if
( n1) movesingle ( start, finish) else
movetower ( n-1, start, temp,
finish) movesingle ( start, finish) movetowe
r ( n-1, temp, finish, start) T(n) 2
T(n-1) 1
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How many steps? disks No. of
steps 1 1 2 3 ( 1 1 1 ) 3 7 ( 3 1
3 ) 4 15 (7 1 7 ) n use recurrence
relation Recurrence relation T(n) 2 T(n 1
) 1
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T(n) 2 T(n 1 ) 1 2 2 T(n 2) 1
1 4 T(n 2) 2 1 4 2 T(n 3) 1
3 8 T(n 3) 7 23 T(n 3) 23 1
2k T(n k) 2k 1 Now let n k
1 2n-1 T(1) 2n-1 1 as time for 1
disk T(1) 1 2. 2n-1 1 2n 1 O(
2n )
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