# Ukkonen's algorithm for longest common substring search

I am working on a programming interview practice question in Python 3 (on codefights.com). The challenge is to return the length of the longest substring that appears in both of two input strings. The code must run in under 4 seconds for all tests, which include a few where the length of the two strings sum to 20,000.

As a shorter example:

longestCommonSubstring("zxabcdezy", "yzabcdezx") -> 6


What can I do to improve my code's runtime?

from math import inf

def longestCommonSubstring(s, t):
"""Return the length of the longest substring that appears in both s and t.
This function builds a suffix tree of a combination of s and t using
Ukkonen's algorithm. It assumes that the symbols $and # appear in neither s nor t. """ len_s = len(s) string = s + '#' + t + '$'
len_string = len(string)
max_len = 0

class LeafNode():

def __init__(self, from_first_word):
self.from_first_word = from_first_word

@property
def has_s_leaves(self):
return self.from_first_word

@property
def has_t_leaves(self):
return not self.from_first_word

class InternalNode():

def __init__(self, root_length):
self.edges = {}  # dictonary of edges keyed by first letter of edge
self.root_length = root_length
self.has_s_leaves = False
self.has_t_leaves = False

def __getitem__(self, key):
return self.edges[key]

def __setitem__(self, key, edge):
self.edges[key] = edge
# Update leaf identity based on new child leaves
# Using "or" is faster than "|=" (I guess |= doesn't short circuit)
self.has_s_leaves = self.has_s_leaves or edge.dest.has_s_leaves
self.has_t_leaves = self.has_t_leaves or edge.dest.has_t_leaves

def __contains__(self, key):
return key in self.edges

class Edge():

def __init__(self, dest, start, end):
self.dest = dest
self.start = start
self.end = end
self.length = self.end - self.start

root = InternalNode(0)

class Cursor():

def __init__(self):
self.node = root
self.edge = None
self.idx = 0
self.lag = -1

def is_followed_by(self, letter):
if self.idx == 0:
return letter in self.node
return letter == string[self.node[self.edge].start + self.idx]

def defer(self, letter):
"""When we defer the insertion of a letter,
we need to advance the cursor one position.
"""
self.idx += 1
# We never want to leave the cursor at the end of an explicit edge.
# If this is the case, move it to the beginning of the next edge.
if self.edge is None:
self.edge = letter
edge = self.node[self.edge]
if self.idx == edge.length:
self.node = edge.dest
self.edge = None
self.idx = 0

def post_insert(self, i):
"""When we are finished inserting a letter, we can pop
it off the front of our queue and prepare the cursor for the
next letter.
"""
self.lag -= 1
# Only when the current node is the root is the first letter (which
# we must remove) part of the cursor edge and index. Otherwise it is
# implicitly determined by the current node.
if self.node is root:
if self.idx > 1:
self.edge = string[i - self.lag]
self.idx -= 1
else:
self.idx = 0
self.edge = None
# Following an insert, we move the active node to the node
# linked from our current active_node or root if there is none.
# When following a suffix link, even to root, it is possible to
# end up with a cursor index that points past the end of the current
# edge. When that happens, follow the edges to a valid cursor
# position. Note that self.idx might be zero and self.edge None.
while self.edge and self.idx >= self.node[self.edge].length:
edge = self.node[self.edge]
self.node = edge.dest
if self.idx == edge.length:
self.idx = 0
self.edge = None
else:
self.idx -= edge.length
self.edge = string[i - self.lag + self.node.root_length]

def split_edge(self):
edge = self.node[self.edge]
# Create a new node and edge
middle_node = InternalNode(self.node.root_length + self.idx)
midpoint = edge.start + self.idx
next_edge = Edge(edge.dest, midpoint, edge.end)
middle_node[string[midpoint]] = next_edge
# Update the current edge to end at the new node
edge.dest = middle_node
edge.end = midpoint
edge.length = midpoint - edge.start
return middle_node

cursor = Cursor()
from_first_word = True
dummy = InternalNode(0)

for i, letter in enumerate(string):

if from_first_word and i > len_s:
from_first_word = False

cursor.lag += 1
prev = dummy  # dummy node to make suffix linking easier the first time

while cursor.lag >= 0:

if cursor.is_followed_by(letter):  # Suffix already exists in tree
cursor.defer(letter)
break

elif cursor.idx != 0:  # We are part-way along an edge
stem = cursor.split_edge()
else:
stem = cursor.node
# Now we have an explicit node and can insert our new edge there.
stem[letter] = Edge(LeafNode(from_first_word), i, inf)
# Whenever we update an internal node, we check for a new max_len
# But not until we have started into the second input string
if (i > len_s and not stem.already_counted
and stem.has_s_leaves and stem.has_t_leaves):
if stem.root_length > max_len:
max_len = stem.root_length
# Link the previously altered internal node to the new node and make
# the new node prev.
cursor.post_insert(i)

return max_len


For sake of comparison, I tested this algorithm against the dynamic programming approach (based on @Ludisposed answer).

Ukkonen's algorithm:

s, t = 1000 * "zxabcdezy", 1000 * "yzabcdezx"
%timeit longestCommonSubstring(s, t)
318 ms ± 12.1 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)


Dynamic programming:

def longestCommonSubstring(S, T):
if len(S) < len(T):
S, T = T, S
row =  * (len(T) + 1)
longest = 0
for i, si in enumerate(S):
for j, tj in enumerate(reversed(T)):
k = len(T) - j
if si == tj:
row[k] = row[k-1] + 1
else:
row[k] = 0
longest = max(longest, *row)
return longest

s, t = 1000 * "zxabcdezy", 1000 * "yzabcdezx"
%timeit longestCommonSubstring(s, t)
27.7 s ± 155 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

• Should not the answer be 6 for the "zxabcdezy" and "yzabcdezx" strings? – alecxe Dec 27 '17 at 3:00
• By the way, longestCommonSubstring(1000 * "zxabcdezy", 1000 * "yzabcdezx") results into a RecursionError: maximum recursion depth exceeded error in the max_common_length() function. Are you sure your solution passes all the tests on codefights? Thanks. – alecxe Dec 27 '17 at 3:06
• @alecxe I ran the code in repl.it with the example provided, and it returned 6 rather than 4, I corrected the post's example case accordingly, must have been a typo. – Phrancis Dec 27 '17 at 4:37
• @Phrancis, thanks for catching my typo. It was supposed to say 6. – Robert Perrotta Dec 27 '17 at 17:49
• @alexce, None of the code fights test cases involved that much repetition (even the long strings). The repetition causes a very deep tree and walking that tree recursively caused the error you found. I changed the code to keep tabs on the max length as it builds the tree rather than walk it recursively after. – Robert Perrotta Dec 27 '17 at 17:51

# Profile

You can use the cProfiler to check where your program takes up the most amount of time

Example Code

import cProfile
pr = cProfile.Profile()
pr.enable()

longestCommonSubstring(s, t)

pr.disable()
pr.print_stats(sort='time')


# DP alternative

There is another yet slower solution, with dynamic programming. However this would take less space but more time $O(n*m)$ where the Suffix tree approach is $O(n + m)$

See the wiki of the Longest Common Substring problem

def longestCommonSubstring(S, T):
L = [*(len(T)+1) for x in range(len(S)+1)]
longest = 0

for i in range(len(S)):
for j in range(len(T)):
if S[i] == T[j]:
if i == 0 or j == 0:
L[i][j] = 1
else:
L[i][j] = L[i-1][j-1] + 1

if L[i][j] > longest:
longest = L[i][j]
else:
L[i][j] = 0
return longest


NOTE that: This solution will not pass all the tests in time.

# Review

Under construction

• (The title mentions Ukkonen, I can't seem to recognise it on codefights.com.) – greybeard Dec 27 '17 at 11:31
• @greybeard codefights.com/interview/Gh37HrvDBrqfLi2iA Challenge link, no mention of Ukkonen – Ludisposed Dec 27 '17 at 16:23
• The dynamic programming approach is too slow. Even with some optimizations to your code that reduce the memory requirements it runs much more slowly than the suffix tree approach (for very long strings). – Robert Perrotta Dec 27 '17 at 17:58
• @RobertPerrotta I see on CF that you have solved this within the time constraints?, feel free to post an answer under your question :) – Ludisposed Dec 28 '17 at 14:19
• @Ludisposed, I finally did solve the challenge! But I wasn't able to optimize my suffix tree method enough to meet the time constraints. Instead, I used a rolling hash/binary search method suggested by some other codefights users. I don't mind sharing my code, but it doesn't really answer the question I asked: how can I make this code run faster? Does it belong here in an answer anyway? – Robert Perrotta Dec 29 '17 at 22:18