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

Which algorithm finds the shortest paths from one source in a graph with non-negative edge weights?

  1. ABellman–Ford is required in all cases
  2. BDepth-first search
  3. CKruskal's algorithm
  4. DDijkstra's algorithm
2.Algorithms & complexityHard

Which statement about little-o notation is correct?

  1. An² = o(n)
  2. B2n = o(n), because 2n grows slowly
  3. Cn = o(n²), but n is not o(n)
  4. Dn = o(n), since every function is little-o of itself
3.Data structuresMedium

An in-order traversal of a binary search tree visits the keys in:

  1. ALevel order
  2. BDescending order always
  3. CInsertion order
  4. DAscending sorted order
4.Data structuresHard

A segment tree is typically used to:

  1. AStore a graph's edges
  2. BAnswer range queries (such as range sums) and updates in O(log n)
  3. CSort strings alphabetically by storing each prefix in a separate tree node
  4. DImplement a stack
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 4 vertices of degree 4, 4 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 34 × deg 44 × deg 53 × deg 2
L =
6.Graph theoryHard
17142031911166ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex F. Click the edge it adds as edge number 6.

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

3Solutions

1.Algorithms & complexity

Answer: D · Dijkstra's algorithm

Why this is the answer

Dijkstra's algorithm works with non-negative weights; Bellman–Ford handles negative weights; Kruskal builds minimum spanning trees.

Answer

D · Dijkstra's algorithm
2.Algorithms & complexity

Answer: C · n = o(n²), but n is not o(n)

Why this is the answer

f = o(g) means f(n) / g(n) tends to 0; n / n² tends to 0, but n / n = 1 does not.

Answer

C · n = o(n²), but n is not o(n)
3.Data structures

Answer: D · Ascending sorted order

Why this is the answer

Visiting the left subtree, then the node, then the right subtree outputs BST keys in sorted order.

Answer

D · Ascending sorted order
4.Data structures

Answer: B · Answer range queries (such as range sums) and updates in O(log n)

Why this is the answer

A segment tree stores aggregates for intervals, supporting range queries and point updates in O(log n) time.

Answer

B · Answer range queries (such as range sums) and updates in O(log n)
5.Graph theory

Answer: L = 24

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 + 48 = 2(L + 13) − 2
L = 24

Answer

L = 24

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

6.Graph theory

Answer: edge AB

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 (11)
2. DE (3)
3. BD (4)
4. BC (1)
5. GD (6)
6. AB (17)

edge number 6

AB

Answer

edge AB

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