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.

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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

Is it correct to say that an O(n²) algorithm is also O(n³)?

  1. ANo, because Big O must always state the exact growth rate of the function
  2. BYes, because Big O is an upper bound and n² grows no faster than n³
  3. CNo, because n³ is smaller than n² for large n
  4. DYes, but only if n is smaller than 100
2.Algorithms & complexityHard

To show that a new problem X is NP-hard, what must you do?

  1. AReduce a known NP-hard problem to X in polynomial time
  2. BShow that X has no polynomial algorithm for small inputs
  3. CReduce X to a known NP-hard problem in polynomial time
  4. DShow that X can be verified in polynomial time
3.Data structuresMedium

A node in a binary search tree has a right subtree. Where is its in-order successor?

  1. AIts right child, always
  2. BThe rightmost node of its left subtree
  3. CThe leftmost node of its right subtree
  4. DIts parent, always
4.Data structuresHard

Which statement about heapsort is correct?

  1. AIt runs in O(n) time on any input
  2. BIt needs a binary search tree as extra storage, so it uses O(n) extra memory
  3. CIt sorts in place in O(n log n) worst-case time but is not stable
  4. DIt is stable and needs O(n) extra memory
5.Graph theoryMedium

A tree has 4 vertices of degree 3, 2 vertices of degree 4, 4 vertices of degree 5, and all its other vertices are leaves (degree 1). How many leaves L does it have?

4 × deg 32 × deg 44 × deg 5
L =
6.Graph theoryHard
10144178212207ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex F. Click the edge it adds as edge number 5.

start: Fk = 5
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge EF
  8. 8edge GB
  9. 9edge GD

3Solutions

1.Algorithms & complexity

Answer: B · Yes, because Big O is an upper bound and n² grows no faster than n³

Why this is the answer

Big O is an upper bound, so the statement is true but not tight; Θ(n²) states the exact growth rate.

Answer

B · Yes, because Big O is an upper bound and n² grows no faster than n³
2.Algorithms & complexity

Answer: A · Reduce a known NP-hard problem to X in polynomial time

Why this is the answer

If a known hard problem can be transformed into X efficiently, a fast algorithm for X would solve the known problem too.

Answer

A · Reduce a known NP-hard problem to X in polynomial time
3.Data structures

Answer: C · The leftmost node of its right subtree

Why this is the answer

All larger keys close to the node are in its right subtree, and the smallest of those is the leftmost node there.

Answer

C · The leftmost node of its right subtree
4.Data structures

Answer: C · It sorts in place in O(n log n) worst-case time but is not stable

Why this is the answer

Heapsort builds a max-heap and extracts the maximum n times within the same array; swaps over long distances can reorder equal keys.

Answer

C · It sorts in place in O(n log n) worst-case time but is not stable
5.Graph theory

Answer: L = 22

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 + 40 = 2(L + 10) − 2
L = 22

Answer

L = 22

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

6.Graph theory

Answer: edge BC

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.

Prim’s algorithm

1. EF (12)
2. DE (2)
3. GD (7)
4. CE (8)
5. BC (4)
6. AB (10)

edge number 5

BC

Answer

edge BC

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