Question: ( 2 5 points ) Consider a two - dimensional array A [ 1 . . n , 1 . . n ] of distinct

(25

points

)

Consider a two

-

dimensional array

A [1 . . n, 1 . . n]

of distinct integers. We want to find the longest increasing path in

A .

A sequence of entries

A [i_{1}, j_{1}], A [i_{2}, j_{2}],

dots,

A [i_{k}, j_{k}], A [i_{k + 1}, j_{k + 1}],

dots is a path in

A

if and only if every two consecutive entries share a common index and the other indices differ by

1,

that is

,

for all

k,

either

i_{k} = i_{k + 1}

and

j_{k + 1} i n {j_{k} - 1, j_{k} + 1},

j_{k} = j_{k + 1}

and

i_{k + 1} i n {i_{k} - 1, i_{k} + 1} .

A path in

A

is increasing

A [i_{1}, j_{1}], A [i_{2}, j_{2}],

dots,

A [i_{k}, j_{k}], A [i_{k + 1}, j_{k + 1}],

dots if and only if

A A [i_{k}, j_{k}] .

The length

o f

a path

i s

the number

o f

entries

i n i t .

Design a dynamic programming algorithm

t o

find the longest increasing path

i n A .

Your algorithm needs

t o

output the maximum length

a s

well

a s

the indices

o f

the array entries

i n

the path. Note that there

i s n o

restriction

o n

where the longest increasing path may start

o r

end. Define and explain your notations. Define and explain your recurrence and boundary conditions. Write your algorithm

i n

pseudo

-

code. Derive the running time

o f

your algorithm.

(25

points

)

Consider a two

-

dimensional array

A [1 . . n, 1 . . n]

of distinct integers. We want to find the longest increasing path in

A .

A sequence of entries

A [i_{1}, j_{1}], A [i_{2}, j_{2}],

dots,

A [i_{k}, j_{k}], A [i_{k + 1}, j_{k + 1}],

dots is a path in

A

if and only if every two consecutive entries share a common index and the other indices differ by

1,

that is

,

for all

k,

either

i_{k} = i_{k + 1}

and

j_{k + 1} i n {j_{k} - 1, j_{k} + 1},

j_{k} = j_{k + 1}

and

i_{k + 1} i n {i_{k} - 1, i_{k} + 1} .

A path in

A

is increasing

A [i_{1}, j_{1}], A [i_{2}, j_{2}],

dots,

A [i_{k}, j_{k}], A [i_{k + 1}, j_{k + 1}],

dots if and only if

A A [i_{k}, j_{k}] .

The length

o f

a path

i s

the number

o f

entries

i n i t .

Design a dynamic programming algorithm

t o

find the longest increasing path

i n A .

Your algorithm needs

t o

output the maximum length

a s

well

a s

the indices

o f

the array entries

i n

the path. Note that there

i s n o

restriction

o n

where the longest increasing path may start

o r

end. Define and explain your notations. Define and explain your recurrence and boundary conditions. Write your algorithm

i n

pseudo

-

code. Derive the running time

o f

your algorithm.

( 2 5 points ) Consider a two - dimensional array

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