More Time-Space Tradeoffs for Finding a Shortest Unique Substring

Round 1

Reviewer 1 Report

The paper presents new results on finding shortest unique substrings. The reasoning builds heavily on earlier results combing them in a clever way.

The paper is not self-contained and it has been written only for experts of the topic. The content is concise without any examples although there is no strict space limit in open-access publishing.

REMARKS

- Your definition of SUS is not precise. I had to look at earlier papers for an exact definition.

- "High probability" in theorems should be specified.

SMALL THINGS

- Line 12: in T

- Fig. 1, line 3: $q$

- Ref. [9] is an M.Sc. thesis.

Author Response

> The paper is not self-contained and it has been written only for experts of the topic. The content is concise without any examples although there is no strict space limit in open-access publishing.

We have added more exposition and a link to Abedin et al.'s recent survey.

> Your definition of SUS is not precise. I had to look at earlier papers for an exact definition.

We have given a more precise definition and examples.

> "High probability" in theorems should be specified.

We have added a restriction that allows us to make "with high probability" mean "the probability of failure is 1 over any given polynomial of n".

> Line 12: in T

Fixed.

> Fig. 1, line 3: $q$

Fixed.

> Ref. [9] is an M.Sc. thesis.

Fixed.

Reviewer 2 Report

please see the attachment

Comments for author File: Comments.pdf

Author Response

> The significance and method of this study are not clearly explained in abstract.
Explain specifically, pls.

We have extended the abstract and tried to make it more informative.

> Generally, when citing references, advice cite them respectively in introduction.

We have included all the citations in the introduction.

Reviewer 3 Report

The paper presents new time-space tradeoffs for the problem of finding shortest unique substring (SUSs) and approximate (k-mismatch) SUSs in a text T, of length n, with alphabet size \sigma.

The authors extend previous results by Senanayaka [9] to find, with high probability, an SUS of length L in O((L/m)n \log L) time using random access over T.
The algorithm is improved by replacing the Karp-Rabin pattern matching used in [9] by a Monte-Carlo randomized sketching method, presented in [13]. The resulting running time is O((L/m)n \log^2(L) \log \log \sigma) taking O((L/m)\log^2 L) sequential passes over T.
The authors also show how to find approximate (k-mismatch) SUSs using results by Gawrychowski and Starikovskaya [14] in O(n^{1+\epsilon}L/m) time and O(n^{\epsilon}L/m) sequential scans over T, for some positive \epsilon and constant k.
In the second part, the paper presents a deterministic algorithm based on DAWG searches [15] for finding (exact) SUSs in O(n\tau \log \sigma \log n) time using O(n/\tau) words of extra space, which improves the best result by Ganguly et al. [8].

There are interesting insights in the paper, the results are sound, and the topic is interesting with applications in Bioinformatics.

I suggest to accept the paper.

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The authors could improve some parts of the manuscript as following.

Major:

1. The background is too short, it requires a good previous knowledge from the reader. I recommend to indicate a textbook for the prerequisites.

2. A table with the summary of results and their theoretical bounds, discussed in Section 1, would be welcome.

3. A pseudo-code (like in Figure 1) for the algorithm presented in Section 4 would be good.

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Minor:

- p2l52: "suffix tree" --> "suffix tree [17-19]" (include citation)
- p2l76: "of the \ell substrings of" --> "of the substrings of"
- p2l77: "lg n" --> "log_2 n"
- p4l129: "qrs" ??
- p4l135: "labeles of" --> "labels of"
- p5l165: "i = [1,2,...,n]" --> "i = 1..n"
- p6l178: "use the the" --> "use the"

Author Response

> 1. The background is too short, it requires a good previous knowledge from the reader. I recommend to indicate a textbook for the prerequisites.

We have extended the introduction and cited Abedin et al.'s recent survey of results on SUSs.

> 2. A table with the summary of results and their theoretical bounds, discussed in Section 1, would be welcome.

Added.

> A pseudo-code (like in Figure 1) for the algorithm presented in Section 4 would be good.

Added.

> p2l52: "suffix tree" --> "suffix tree [17-19]" (include citation)

Fixed.

> p2l76: "of the \ell substrings of" --> "of the substrings of"

Fixed.

> p2l77: "lg n" --> "log_2 n"

Fixed.

> p4l129: "qrs" ??

Fixed (changed to "q r s").

> p4l135: "labeles of" --> "labels of"

Fixed.

> p5l165: "i = [1,2,...,n]" --> "i = 1..n"

Fixed.

> p6l178: "use the the" --> "use the"

Fixed.