Miss Samaraweera was last seen at a Somerfield shop at about 0100 BST
Officers looking for a murderer and suspected serial rapist are asking men in east London to give DNA samples.
Detectives believe whoever raped and strangled Michelle Samaraweera, 35, in Walthamstow, lives nearby.
Police believe the same man may have raped two other women - a 59-year-old in March and a 46-year-old in April, both less than a mile from the murder.
House-to-house inquiries have begun with those matching the suspect's description asked to provide DNA.
The suspect is described as Asian, or with dark skin, and aged between 30 and 45 years.
...this is the quickest and most comprehensive way to eliminate men from our investigation
Det Supt Vic Rae
Miss Samaraweera's partially-clothed body was found by a passer-by in Queens Road Park at 0550 BST on 30 May.
She was last seen alive at a nearby Somerfield shop five hours earlier. Officers are appealing for help in trying to trace her movements from 2200 BST on 29 May to when her body was found.
Police said she was a widower who was living with her boyfriend in Walthamstow. She had no children.
Det Supt Vic Rae said: "The DNA test is designed solely to apprehend the individual responsible for Michelle's brutal murder and the two linked sexual assaults.
"Much consideration and planning has gone into this decision and we feel this is the quickest and most comprehensive way to eliminate men from our investigation."
Terrible ordeal
He added: "While stranger murders and sexual assaults are incredibly rare we have to be mindful of the terrible ordeal these ladies endured.
"Michelle suffered a terrifying assault and the physical abuse against the other two women was significant."
Her murder has been linked with two similar attacks.
The first took place at an address close to Queens Road on 24 March when a 59-year-old woman was forced inside her home and raped.
The second, in which a 46-year-old woman was attacked in an alleyway off nearby Lea Bridge Road, took place in the early hours of 22 April.
The incidents have led to fears in the area, with businesses offering female employees taxi rides home and pubs putting up signs to warn people to be on their guard late at night.
Waltham Forest Council will hold a meeting with police on Tuesday to discuss what other precautions can be taken.
Two men arrested on 30 May have been released with no further action.
A further two men arrested the next day have been bailed until July pending further inquiries.
A Metropolitan Police spokesman said he believed the last time they carried a similar DNA sweep was is 2006 for the Sally Anne Bowman murder investigation in South Croydon, south London.
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