Revisiting Sparse Dynamic Programming for the 0/1 Knapsack ...Sparse DP Algorithm for solving...

Revisiting Sparse Dynamic

Programming for the 0/1 Knapsack

Problem

Tarequl Islam Sifat

[email protected]

Corespeq Inc

Fort Collins, Colorado, USA

Nirmal Prajapati

[email protected]

Los Alamos National Laboratory

Los Alamos, New Mexico, USA

Sanjay Rajopadhye

[email protected]

Department of Computer Science

Colorado State University

Fort Collins, Colorado, USA

0/1 Knapsack Problem Statement

◼ Given a set of 𝑁 items numbered from 1 up to 𝑁, each with a weight

𝑤𝑖 and a profit 𝑝𝑖, along with maximum capacity 𝐶, we must

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒

𝑖=1

𝑁+1

𝑝𝑖𝑥𝑖

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

𝑖=1

𝑁+1

𝑤𝑖𝑥𝑖 < 𝐶 𝑎𝑛𝑑 𝑥𝑖 ∈ {0,1}

2

Sparse DP Algorithm for solving 0/1-

Knapsack Problem◼ A “sparse” KPDP algorithm (SKPDP) has been known for a while.

◼ Conventional KPDP algorithm generates a DP table that contains

many repeated values.

◼ SKPDP does not calculate the repeated values in the DP table.

◼ So far there has been no quantitative analysis of its benefits.

3

Contributions

◼ Quantitative analysis of Sequential SKPDP

◼ Exploration of two parallelization techniques for SKPDP and their

performance analysis

◼ Comparison of SKPDP with Branch-and-Bound

4

Problem Instance Generation

For Quantitative Analysis◼ Uncorrelated : 𝑝𝑖 = 𝑟𝑎𝑛𝑑𝑜𝑚(𝑟𝑎𝑛𝑔𝑒)

◼ Weakly Correlated : 𝑝𝑖 = 𝛼𝑤𝑖 + 𝑟𝑎𝑛𝑑𝑜𝑚(𝑟𝑎𝑛𝑔𝑒)

◼ Strongly Correlated: 𝑝𝑖 = 𝛼𝑤𝑖 + 𝛽 [Hardest Problem Instances]

◼ Subset Sum: 𝑝𝑖 = 𝑤𝑖

where 𝛼, 𝛽 are constants, 𝑝𝑖 is profit and 𝑤𝑖 is weight of each item.

5

Punch Line

◼ For KP instances with significantly large capacity than the number of

items (C >> N)

◼ If the problem instance is weakly correlated, the operation count of

SKPDP algorithm is invariant with respect to capacity (𝐶).

◼ If the problem instance is strongly correlated, the operation count of

SKPDP algorithm is exponentially less than KPDP.

6

Punch Line

7

Weakly Correlated Instances Strongly Correlated Instances

Dynamic Programming Solution

8

for ( k=1 ; k < N ; k++ ) {for ( c=0 ; c <= C ; c++ ) {

if( c < weights[k] )M[k,c] = M[k-1,c];

elseM[k,c]=MAX(M[k-1,c], M[k-1,c-weights[k]]+profits[k]);

}}

𝑀 𝑘, 𝑐 = ቊ𝑀 𝑘 − 1, 𝑐 𝑖𝑓 𝑤𝑘 > 𝑐

max 𝑀 𝑘 − 1, 𝑐 ,𝑀 𝑘 − 1, 𝑐 − 𝑤𝑘 + 𝑝𝑘 𝑒𝑙𝑠𝑒

Example Problem Instance

Item No. Profits Weights

1 1 1

2 6 2

3 18 5

4 22 6

5 28 7

9

𝑁 = 5 𝐶 = 11

Dynamic Programming Table

10

𝑀 𝑘, 𝑐 = ቊ𝑀 𝑘 − 1, 𝑐 𝑖𝑓 𝑤𝑘 > 𝑐

max 𝑀 𝑘 − 1, 𝑐 ,𝑀 𝑘 − 1, 𝑐 − 𝑤𝑘 + 𝑝𝑘 𝑒𝑙𝑠𝑒

Capacity

Item# (weight,profit) 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 (1,1) 0 1 1 1 1 1 1 1 1 1 1 1

2 (2,6) 0 1 6 7 7 7 7 7 7 7 7 7

3 (5,18) 0 1 6 7 7 18 19 24 25 25 25 25

4 (6,22) 0 1 6 7 7 18 22 24 28 29 29 40

5 (7,28) 0 1 6 7 7 18 22 28 29 34 35 40

c

k

Memory Efficient KPDP

◼ Only need the current row of the DP table to calculate the next

row

◼ The whole table does not have to be stored

◼ This way we can find the optimal profit value

◼ We can find the exact solution including which items are taken in

the optimal solution by using a divide-and-conquer strategy

◼ Divide-and-Conquer strategy doubles the number of

computations, 2𝑁𝐶

◼ Reduces memory requirement by a factor of 𝑁/2

11

“Sparsity” in the current context

12

“Sparsity” in the current context

Capacity

Item# (weight,profit) 0 1 2 3 4 5 6 7 8 9 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0

1 (1,1) 0 1 1 1 1 1 1 1 1 1 1 1

2 (2,6) 0 1 6 7 7 7 7 7 7 7 7 7

3 (5,18) 0 1 6 7 7 18 19 24 25 25 25 25

4 (6,22) 0 1 6 7 7 18 22 24 28 29 29 40

5 (7,28) 0 1 6 7 7 18 22 28 29 34 35 40

13

Item#

0 <0,0>

1 (1,1) <0,0>,<1,1>

2 (2,6) <0,0>,<1,1>,<2,6>,<3,7>

3 (5,18) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>

4 (6,22) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,22>,<7,24>,<8,28>,<9,29>,<11,40>

5 (7,28) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,22>,<7,28>,<8,29>,<9,34>,<10,35>,<11,40>

Building the Sparse Table

Add-Merge-Kill

14

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>

When we include the 4th item (Weight: 6, Profit: 22) in our choice

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>


Add-Merge-Kill

14

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>


<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>


Add-Merge-Kill

14

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>


Add (6,22) to each pair,

<6,22>,<7,23>,<8,28>,<9,29>,<11,40>,<12,41>,<13,46>,<14,47>

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>


Add-Merge-Kill

14

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>


Add (6,22) to each pair,

<6,22>,<7,23>,<8,28>,<9,29>,<11,40>

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>


Add-Merge-Kill

14

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>


<6,22>,<7,23>,<8,28>,<9,29>,<11,40>


Add-Merge-Kill

15

Generation of Problem Instances

◼ The fraction of objects that can fit in the knapsack on average, 1/𝜆

𝑊𝑎𝑣𝑔 =𝜆𝐶

𝑁

◼ The set of weights, 𝑤𝑖 is generated with a normal distribution that

has a mean of 𝑊𝑎𝑣𝑔

◼ For weakly correlated problem instances, the correlation between

the weights and profits is controlled by a noise factor, 𝜎𝑝𝑖 = 𝛼𝑤𝑖 + 𝑅𝑎𝑛𝑑𝑜𝑚 𝐼𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 [−𝜎𝑊𝑎𝑣𝑔, 𝜎𝑊𝑎𝑣𝑔]

◼ For strongly correlated problem instances 𝜎 is irrelevant,

𝑝𝑖 = 𝛼𝑤𝑖 + 𝛽

16

Gain

𝐺𝑎𝑖𝑛 = 1 −𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑆𝐾𝑃𝐷𝑃

𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝐾𝑃𝐷𝑃 (= 2𝑁𝐶)

The range of Gain is (-1, 1)

◼ A value of gain close to 1 means that the number of iterations

in SKPDP is insignificant compared to KPDP

◼ A value of gain close to -1 mean that we have the worst-case

scenario for SKPDP.

17

Gain

18

𝜆 = 2, 𝜎 = 0.1%

SKPDP vs KPDP

19

Weakly Correlated Instances Strongly Correlated Instances

𝜆 = 2, 𝜎 = 0.1% 𝑁 = 256, 𝜆 = 8, 𝜎 = 0.1%

Impact of 𝜎 on the sparsity

20

𝑝 = 𝛼𝑤 + 𝑅𝑎𝑛𝑑𝑜𝑚 𝐼𝑛𝑡. 𝐼𝑛 −𝜎𝑊𝑎𝑣𝑔, 𝜎𝑊𝑎𝑣𝑔 ; 𝜆 = 2

Impact of 𝜆 on the sparsity

21

𝑊𝑎𝑣𝑔 =𝜆𝐶

𝑁; 𝜎 = 0.1%

Impact of 𝜎 and 𝜆 on the sparsity

22

𝑁 = 210, 212 < C < 250

Parallelization of SKPDP

◼ Fine-Grained Parallelization

◼ Coarse-Grained Parallelization

23

Fine-Grained Parallelization

24


Item#

0 <0,0>

1 (1,1) <0,0>,<1,1>

2 (2,6) <0,0>,<1,1>,<2,6>,<3,7>

3 (5,18) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>

4 (6,22) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,22>,<7,24>,<8,28>,<9,29>,<11,40>

5 (7,28) <0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,22>,<7,28>,<8,29>,<9,34>,<10,35>,<11,40>

25

{1,2,3} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,19>, <7,24>, <8,25>

{1,2,3,4} <0,0>, <1,1>, <2,6>, <3,7>, <5,18>, <6,22>, <7,24>, <8,28>, <9,29>, <11,40>


Remember the example used to demonstrate Add-Merge-Kill?


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>

<0,0>,<1,1>,<2,6>,<3,7>,<6,22>,<7,23>,<8,28>,<9,29>

<5,18>,<6,19>,<7,24>,<8,25>,<11,40>


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>

<0,0>,<1,1>,<2,6>,<3,7>,<6,22>,<7,23>,<8,28>,<9,29>

<5,18>,<6,19>,<7,24>,<8,25>,<11,40>

At most 𝑤𝑖 pairs from the suffix of the preceding part


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<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>

<0,0>,<1,1>,<2,6>,<3,7>,<6,22>,<7,23>,<8,28>,<9,29>

<5,18>,<6,19>,<7,24>,<8,25>,<11,40>


At most 𝑤𝑖 pairs from the prefix of the following part


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>

<0,0>,<1,1>,<2,6>,<3,7>,<6,22>,<7,23>,<8,28>,<9,29>

<5,18>,<6,19>,<7,24>,<8,25>,<11,40>


At most 𝑤𝑖 pairs from the prefix of the following part

In the worst-case extra merge-kill we must do at each point of stitching is 2𝑤𝑖


26

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4th row:

<5,18>,<6,19>,<7,24>,<8,25><0,0>,<1,1>,<2,6>,<3,7>

<0,0>,<1,1>,<2,6>,<3,7>,<6,22>,<7,23>,<8,28>,<9,29>

<5,18>,<6,19>,<7,24>,<8,25>,<11,40>

<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,22>,<7,24>,<8,28>,<9,29>,<11,40>

Coarse-Grained Parallelization

27


◼ The computation occurs in batches of P rows.

◼ Every 𝑃𝑡ℎ row of the DP table is stored (completely) in the

memory.

◼ Other 𝑃 − 1 rows only stores a local buffer of size 𝑤𝑖

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<0,0>,<1,1>,<2,6>,<3,7>,<5,18>,<6,19>,<7,24>,<8,25>4rd row:

k

j

At most 𝑤𝑖


◼ The local buffers are basically a FIFO stack and implemented

with a rotating array.

◼ These rotating arrays have a length of 𝑊𝑚𝑎𝑥 = max(𝑤𝑖)

◼ Extra memory required by the implementation is 𝑃 ∗𝑊𝑚𝑎𝑥

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Performance of Coarse-Grained

Parallelization

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𝜆 = 8, 𝜎 = 0.1%

SKPDP vs BnB

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Future work

◼ A model to predict the performance gain by SKPDP

◼ Ordering the items by weight values to maximize sparsity

◼ Choosing the best input parameters for the coarse-grained

parallelization of SKPDP

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Conclusion

◼ SKPDP shows exponential performance improvement using

SKPDP over KPDP.

◼ Coarse-grained parallelization shows promise.

◼ For most problem instances BnB with proper heuristics does

better than SKPDP.

◼ For some hard problem instances SKPDP performs better

than BnB.

33

34

Thank you!

Revisiting Sparse Dynamic Programming for the 0/1 Knapsack ...Sparse DP Algorithm for solving...

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