Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

New problemsBack to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

With coins of 1, 3 and 4 and an amount of 6, how many coins does the greedy "largest coin first" method use, compared to the optimum?

  1. AGreedy uses 4 coins; the optimum is 3
  2. BGreedy uses 3 coins; the optimum is 3
  3. CGreedy uses 2 coins; the optimum is 2
  4. DGreedy uses 3 coins; the optimum is 2
2.Algorithms & complexityHard

Insertion sort runs in O(n) on sorted input. Does this contradict the Ω(n log n) lower bound for comparison sorting?

  1. ANo, because insertion sort is not a comparison sort
  2. BYes, the lower bound is therefore wrong
  3. CNo, the lower bound applies to the worst case over all inputs, not to every input
  4. DYes, but only for arrays shorter than 100
3.Data structuresMedium

How long does a search for a word of length L take in a trie with a fixed alphabet?

  1. AO(log n), where n is the number of words
  2. BO(n), where n is the number of words
  3. CO(L × n)
  4. DO(L), independent of how many words are stored
4.Data structuresHard

How can you detect whether a singly linked list contains a cycle using O(1) extra memory?

  1. AStore every visited node in a hash set
  2. BSort the list and look for duplicates
  3. CMove one pointer by one step and another by two steps and check whether they meet
  4. DCount the nodes until the count exceeds the list length field
5.Graph theoryMedium

A tree has 2 vertices of degree 5, 3 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

2 × deg 53 × deg 2
L =
6.Graph theoryHard
11312101415261315ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: D · Greedy uses 3 coins; the optimum is 2

Why this is the answer

Greedy takes 4 + 1 + 1 = 3 coins, but 3 + 3 uses only 2, so greedy is not optimal for this coin system; dynamic programming is.

Answer

D · Greedy uses 3 coins; the optimum is 2
2.Algorithms & complexity

Answer: C · No, the lower bound applies to the worst case over all inputs, not to every input

Why this is the answer

The bound says every comparison sort needs about n log n comparisons on some input of size n; special inputs can be much faster.

Answer

C · No, the lower bound applies to the worst case over all inputs, not to every input
3.Data structures

Answer: D · O(L), independent of how many words are stored

Why this is the answer

Each character selects one child, so a lookup takes L steps no matter how many words the trie contains.

Answer

D · O(L), independent of how many words are stored
4.Data structures

Answer: C · Move one pointer by one step and another by two steps and check whether they meet

Why this is the answer

In Floyd's algorithm, if there is a cycle the fast pointer eventually catches up with the slow one; otherwise the fast pointer reaches null.

Answer

C · Move one pointer by one step and another by two steps and check whether they meet
5.Graph theory

Answer: L = 8

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Handshake lemma for a tree

L + 16 = 2(L + 5) − 2
L = 8

Answer

L = 8

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: d(F) = 26

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 11, C: 3
C (3): G: 9
G (9): D: 22, E: 24
B (11): D: 21
D (21): F: 36
E (24): F: 26
F (26)

Shortest path

A → C → G → E → F
d(F) = 26

Answer

d(F) = 26

ExamProblems · examproblems.com/courses/data-structures-and-algorithms