(algorithm)

**Definition:**
*Merge* n sorted sequences of different lengths into one output while minimizing reads. Only two sequences can be merged at once. At each step, the two shortest sequences are merged.

**Formal Definition:** Let D={n_{1}, … , n_{k}} be the set of lengths of sequences to be merged. Take the two shortest sequences, n_{i}, n_{j}∈ D, such that n≥ n_{i} and n≥ n_{j} ∀ n∈ D. Merge these two sequences. The new set D is D' = (D - {n_{i}, n_{j}}) ∪ {n_{i}+n_{j}}. Repeat until there is only one sequence.

**See also**
*simple merge*, *ideal merge*, *Huffman coding*, *greedy algorithm*.

*Note:
Merging sequences by length is the same as joining trees by frequency in Huffman coding. For example, let there be a set of sorted sequences of the following lengths: D={3,5,7,9,12,14,15,17}. Building the optimal merge tree goes as follows. Note that merged sequences are replaced by the sum of their lengths. For instance, the first step merges the sequence of length 3 and the sequence of length 5 to get a sequence of length 8.
*

3 5 7 9 12 14 15 17

8 7 9 12 14 15 17

/ \

3 5

15 9 12 14 15 17

/ \

8 7

/ \

3 5

15 21 14 15 17

/ \ / \

8 7 9 12

/ \

3 5

29 21 15 17

/ \ / \

14 15 9 12

/ \

8 7

/ \

3 5

29 21 32

/ \ / \ / \

14 15 9 12 15 17

/ \

8 7

/ \

3 5

50 32

/ \ / \

/ \ 15 17

29 21

/ \ / \

14 15 9 12

/ \

8 7

/ \

3 5

82

/ \

/ \

/ \

50 32

/ \ / \

/ \ 15 17

29 21

/ \ / \

14 15 9 12

/ \

8 7

/ \

3 5

Author: AL

Go to the Dictionary of Algorithms and Data Structures home page.

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Entry modified 14 August 2008.

HTML page formatted Wed Mar 13 12:42:46 2019.

Cite this as:

Alen Lovrencic, "optimal merge", in
*Dictionary of Algorithms and Data Structures* [online], Paul E. Black, ed. 14 August 2008. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/optimalMerge.html