Suffix arrays in Data Structures Theory - Step-by-Step Execution

Concept Flow - Suffix arrays

Input String S

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Generate all suffixes of S

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Sort suffixes lexicographically

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Store starting indices of sorted suffixes

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Suffix Array constructed

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Use suffix array for fast substring queries

Start with the input string, list all suffixes, sort them alphabetically, then record their starting positions to form the suffix array.

Execution Sample

Data Structures Theory

S = "banana"
suffixes = [(S[i:], i) for i in range(len(S))]
suffixes.sort()
suffix_array = [pos for (suf, pos) in suffixes]
print(suffix_array)

This code creates all suffixes of "banana", sorts them, and prints the array of starting indices in sorted order.

Analysis Table

Step	Operation	Suffix Generated	Suffix Index	Suffixes Sorted So Far	Suffix Array State
1	Generate suffix	banana	0	[('banana',0)]	[]
2	Generate suffix	anana	1	[('banana',0), ('anana',1)]	[]
3	Generate suffix	nana	2	[('banana',0), ('anana',1), ('nana',2)]	[]
4	Generate suffix	ana	3	[('banana',0), ('anana',1), ('nana',2), ('ana',3)]	[]
5	Generate suffix	na	4	[('banana',0), ('anana',1), ('nana',2), ('ana',3), ('na',4)]	[]
6	Generate suffix	a	5	[('banana',0), ('anana',1), ('nana',2), ('ana',3), ('na',4), ('a',5)]	[]
7	Sort suffixes lex order	-	-	[('a',5), ('ana',3), ('anana',1), ('banana',0), ('na',4), ('nana',2)]	[]
8	Extract indices	-	-	-	[5, 3, 1, 0, 4, 2]
9	Print suffix array	-	-	-	[5, 3, 1, 0, 4, 2]

💡 All suffixes generated and sorted; suffix array completed with starting indices.

State Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	After Step 8	Final
S	"banana"	"banana"	"banana"	"banana"	"banana"	"banana"	"banana"	"banana"	"banana"	"banana"
suffixes	[]	[('banana',0)]	[('banana',0), ('anana',1)]	[('banana',0), ('anana',1), ('nana',2)]	[('banana',0), ('anana',1), ('nana',2), ('ana',3)]	[('banana',0), ('anana',1), ('nana',2), ('ana',3), ('na',4)]	[('banana',0), ('anana',1), ('nana',2), ('ana',3), ('na',4), ('a',5)]	[('a',5), ('ana',3), ('anana',1), ('banana',0), ('na',4), ('nana',2)]	[('a',5), ('ana',3), ('anana',1), ('banana',0), ('na',4), ('nana',2)]	[('a',5), ('ana',3), ('anana',1), ('banana',0), ('na',4), ('nana',2)]
suffix_array	[]	[]	[]	[]	[]	[]	[]	[]	[5, 3, 1, 0, 4, 2]	[5, 3, 1, 0, 4, 2]

Key Insights - 3 Insights

Why do we store the starting indices of suffixes instead of the suffix strings themselves?

Storing indices saves space and allows quick reference to the original string. The execution_table row 8 shows extracting indices after sorting suffixes.

Why must suffixes be sorted lexicographically?

Sorting suffixes alphabetically arranges all substrings in order, enabling fast search. See execution_table row 7 where suffixes are sorted.

How does the suffix array help in substring search?

Because suffixes are sorted, binary search on suffix_array indices quickly finds substrings. This is implied after the suffix array is built in execution_table row 9.

Visual Quiz - 3 Questions

Look at the execution_table at step 7. Which suffix is first in the sorted list?

A "banana" starting at index 0

B "a" starting at index 5

C "ana" starting at index 3

D "nana" starting at index 2

💡 Hint

Check the 'Suffixes Sorted So Far' column at step 7 in execution_table.

At which step does the suffix array get its final values?

A Step 8

B Step 7

C Step 6

D Step 9

💡 Hint

Look at the 'Suffix Array State' column in execution_table to see when indices are extracted.

If the input string was "apple", how many suffixes would be generated?

A 4

B 6

C 5

D 7

💡 Hint

Refer to variable_tracker 'suffixes' length after all suffixes generated for "banana" which has length 6.

Concept Snapshot

Suffix arrays list all suffixes of a string.
They are sorted alphabetically.
We store the starting indices of these sorted suffixes.
This structure helps fast substring search.
Building suffix arrays involves generating suffixes, sorting, and indexing.

Full Transcript

Suffix arrays are built by taking all suffixes of a string, sorting them alphabetically, and recording their starting positions. For example, for the string 'banana', we generate suffixes like 'banana', 'anana', 'nana', 'ana', 'na', and 'a'. These suffixes are sorted lexicographically to get 'a', 'ana', 'anana', 'banana', 'na', 'nana'. We then create the suffix array by storing the starting indices of these sorted suffixes, resulting in [5, 3, 1, 0, 4, 2]. This array allows efficient substring searches by binary searching the suffixes. The process involves generating suffixes, sorting them, and extracting indices, as shown step-by-step in the execution table and variable tracker. Key points include why indices are stored instead of strings, the importance of sorting, and how the suffix array supports fast queries.