Monthly Archives: November 2014

math teaching

Compactness and Open Covers


Let \(S \subseteq \mathbf{R}^n\) and suppose \(\mathcal{A} = \{A_i | i \in I\}\) is a family of open subsets of \(\mathbf{R}^n\) such that
\[
S \subseteq \bigcup_{i \in I} A_i.
\]
We call such a family \(\mathcal{A}\) an open cover of \(S\). We call a subset \(K \subseteq \mathbf{R}^n\) topologically compact if every open cover \(\mathcal{A}\) of \(K\) admits a finite subcover. That is for every open cover \(\mathcal{A}\), there exists \(k \in \mathbf{N}\) such that there exist \(A_1, A_2, \ldots, A_k \in \mathcal{A}\) with
\[
K \subseteq \bigcup_{i = 1}^k A_i.
\]

Theorem. If \(K \subseteq \mathbf{R}^n\) is topologically compact, then it is closed and bounded.

Proof. We first show that \(K\) is bounded. To this end, consider the family of open sets given by \(A_i = \{x \in \mathbf{R}^n \big| |x| < i\}\). Clearly, we have \[ K \subseteq \mathbf{R}^n = \bigcup_{i = 1}^\infty A_i. \] Since \(K\) admits a finite subcover, there exist \(i_1 < i_2 < \cdots < i_k\) such that \[ K \subseteq A_{i_1} \cup A_{i_2} \cup \cdots \cup A_{i_k}. \] Since we have \(A_i \subseteq A_j\) for \(i < j\), we in fact have \(K \subseteq A_{i_k}\), hence \(K\) is bounded. We will now argue that \(K\) is closed. Suppose toward a contradiction that \(K\) is not closed. Then there exists some limit point \(x\) of \(K\) which is not contained in \(K\). Now consider the family of sets \[ A_i = \{y \in \mathbf{R}^n \big| |x - y| > 1/i\}.
\]
Note that \(\bigcup A_i = \mathrm{R} \setminus\{x\}\), thus (since \(x \notin K\)) \(\{A_i\}\) is an open cover of \(K\). Since this family admits an open subcover and the \(A_i\) are nested (\(i < j\) implies \(A_i \subseteq A_j\)) we have, in fact, \(K \subseteq A_k\) for some \(k \in \mathbf{N}\). But this implies that \(B(x, 1/k) \subseteq K^c\), contradicting the assumption that \(x\) is a limit point of \(K\).∎ The converse of the theorem above is also true. The equivalence of being topologically compact and being closed and bounded (in \(\mathbf{R}^n\)) is known as the Heine-Borel Theorem.

computer science expository math

Shorter Certificates for Set Disjointness


Suppose two players, Alice and Bob, each hold equal sized subsets of \([n] = \{1, 2, \ldots, n\}\). A third party, Carole, wishes to convince Alice and Bob that their subsets are disjoint, i.e., their sets have no elements in common. How efficiently can Carole prove to Alice and Bob that their sets are disjoint?

To formalize the situation, suppose \(k\) is an integer with \(1 < k < n/2\). Alice holds a set \(A \subseteq [n]\) with \(|A| = k\) and similarly Bob holds \(B \subseteq [n]\) with \(|B| = k\). We assume that Carole sees the sets \(A\) and \(B\), but Alice and Bob have no information about each others' sets. We allow Carole to send messages to Alice and Bob, and we allow Alice and Bob to communicate with each other. Our goal is to minimize the total amount of communication between Carole, Alice, and Bob. In particular, we will consider communication protocols of the following form:

  1. Carole produces a certificate or proof that \(A \cap B = \emptyset\) and sends this certificate to Alice and Bob.
  2. Alice and Bob individually verify that Carole’s certificate is valid with respect to their individual inputs. Alice and Bob then send each other (very short) messages saying whether or not they accept Carole’s proof. If they both accept the proof, they can be certain that their sets are disjoint
  3. In order for the proof system described above to be valid, it must satisfy the following two properties:

    Completeness If \(A \cap B = \emptyset\) then Carole can send a certificate that Alice and Bob both accept.

    Soundness If \(A \cap B \neq \emptyset\) then any certificate that Carole sends must be rejected by at least one of Alice and Bob.

    Before giving a “clever” solution to this communication problem, we describe a naive solution. Since Carole sees \(A\) and \(B\), her proof of their disjointness could simply be to send Alice and Bob the (disjoint) pair \(C = (A, B)\). Then Alice verifies the validity of the certificate \(C\) by checking that her input \(A\) is equal to the first term in \(C\), and similarly Bob checks that \(B\) is equal to the second term. Clearly, if \(A\) and \(B\) are disjoint, Alice and Bob will both accept \(C\), while if \(A\) and \(B\) intersect, Alice or Bob will reject every certificate that Carole could send.

    Let us quickly analyze the efficiency of this protocol. The certificate that Carole sends consists of a pair of \(k\)-subsets of \([n]\). The naive encoding of simply listing the elements of \(A\) and \(B\) requires \(2 k \log n\) bits — each \(i \in A \cup B\) requires \(\log n\) bits, and there are \(2 k\) such indices in the list. In fact, even if Carole is much cleverer in her encoding of the sets \(A\) and \(B\), she cannot compress the proof significantly for information theoretic reasons. Indeed, there are
    $$
    {n \choose k}{n-k \choose k}
    $$
    distinct certificates that Carole must be able to send, hence her message must be of length at least
    $$
    \log {n \choose k} \geq \log ((n/k)^k) = k \log n – k \log k.
    $$
    Is it possible for Carole, Alice, and Bob to devise a more efficient proof system for disjointness?

    read more »