Fingerprint-based　positioning　in　a　wireless　local　area　network　(WLAN)　environment　has　received　much　attention　recently.　One　key　issue　for　the　positioning　method　is　the　radio　map　construction,　which　generally　requires　significant　effort　to　collect　enough　measurements　of　received　signal　strength　(RSS).　Based　on　the　observation　that　RSSs　have　high　spatial　correlation,　we　propose　an　efficient　radio　map　construction　method　based　on　low-rank　approximation.　Different　from　the　conventional　interpolation　methods,　the　proposed　method　represents　the　distribution　of　RSSs　as　a　low-rank　matrix　and　constructs　the　dense　radio　map　from　relative　sparse　measurements　by　a　revised　low-rank　matrix　completion　method.　To　evaluate　the　proposed　method,　both　simulation　tests　and　field　experiments　have　been　conducted.　The　experimental　results　indicate　that　the　proposed　method　can　reduce　the　RSS　measurements　evidently.　Moreover,　using　the　constructed　radio　maps　for　positioning,　the　positioning　accuracy　is　also　improved.